1 /*
  2     Copyright 2008-2017
  3         Matthias Ehmann,
  4         Michael Gerhaeuser,
  5         Carsten Miller,
  6         Bianca Valentin,
  7         Alfred Wassermann,
  8         Peter Wilfahrt
  9 
 10     This file is part of JSXGraph.
 11 
 12     JSXGraph is free software dual licensed under the GNU LGPL or MIT License.
 13 
 14     You can redistribute it and/or modify it under the terms of the
 15 
 16       * GNU Lesser General Public License as published by
 17         the Free Software Foundation, either version 3 of the License, or
 18         (at your option) any later version
 19       OR
 20       * MIT License: https://github.com/jsxgraph/jsxgraph/blob/master/LICENSE.MIT
 21 
 22     JSXGraph is distributed in the hope that it will be useful,
 23     but WITHOUT ANY WARRANTY; without even the implied warranty of
 24     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 25     GNU Lesser General Public License for more details.
 26 
 27     You should have received a copy of the GNU Lesser General Public License and
 28     the MIT License along with JSXGraph. If not, see <http://www.gnu.org/licenses/>
 29     and <http://opensource.org/licenses/MIT/>.
 30  */
 31 
 32 
 33 /*global JXG: true, define: true*/
 34 /*jslint nomen: true, plusplus: true*/
 35 
 36 /* depends:
 37  jxg
 38  math/math
 39  math/geometry
 40  math/numerics
 41  math/statistics
 42  base/constants
 43  base/coords
 44  base/element
 45  utils/type
 46   elements:
 47    transform
 48    point
 49    ticks
 50  */
 51 
 52 /**
 53  * @fileoverview The geometry object Line is defined in this file. Line stores all
 54  * style and functional properties that are required to draw and move a line on
 55  * a board.
 56  */
 57 
 58 define([
 59     'jxg', 'math/math', 'math/geometry', 'math/numerics', 'math/statistics', 'base/constants', 'base/coords',
 60     'base/element', 'utils/type', 'base/point'
 61 ], function (JXG, Mat, Geometry, Numerics, Statistics, Const, Coords, GeometryElement, Type, Point) {
 62 
 63     "use strict";
 64 
 65     /**
 66      * The Line class is a basic class for all kind of line objects, e.g. line, arrow, and axis. It is usually defined by two points and can
 67      * be intersected with some other geometry elements.
 68      * @class Creates a new basic line object. Do not use this constructor to create a line.
 69      * Use {@link JXG.Board#create} with
 70      * type {@link Line}, {@link Arrow}, or {@link Axis} instead.
 71      * @constructor
 72      * @augments JXG.GeometryElement
 73      * @param {String,JXG.Board} board The board the new line is drawn on.
 74      * @param {Point} p1 Startpoint of the line.
 75      * @param {Point} p2 Endpoint of the line.
 76      * @param {String} id Unique identifier for this object. If null or an empty string is given,
 77      * an unique id will be generated by Board
 78      * @param {String} name Not necessarily unique name. If null or an
 79      * empty string is given, an unique name will be generated.
 80      * @param {Boolean} withLabel construct label, yes/no
 81      * @param {Number} layer display layer [0-9]
 82      * @see JXG.Board#generateName
 83      */
 84     JXG.Line = function (board, p1, p2, attributes) {
 85         this.constructor(board, attributes, Const.OBJECT_TYPE_LINE, Const.OBJECT_CLASS_LINE);
 86 
 87         /**
 88          * Startpoint of the line. You really should not set this field directly as it may break JSXGraph's
 89          * udpate system so your construction won't be updated properly.
 90          * @type JXG.Point
 91          */
 92         this.point1 = this.board.select(p1);
 93 
 94         /**
 95          * Endpoint of the line. Just like {@link JXG.Line.point1} you shouldn't write this field directly.
 96          * @type JXG.Point
 97          */
 98         this.point2 = this.board.select(p2);
 99 
100         /**
101          * Array of ticks storing all the ticks on this line. Do not set this field directly and use
102          * {@link JXG.Line#addTicks} and {@link JXG.Line#removeTicks} to add and remove ticks to and from the line.
103          * @type Array
104          * @see JXG.Ticks
105          */
106         this.ticks = [];
107 
108         /**
109          * Reference of the ticks created automatically when constructing an axis.
110          * @type JXG.Ticks
111          * @see JXG.Ticks
112          */
113         this.defaultTicks = null;
114 
115         /**
116          * If the line is the border of a polygon, the polygon object is stored, otherwise null.
117          * @type JXG.Polygon
118          * @default null
119          * @private
120          */
121         this.parentPolygon = null;
122 
123         /* Register line at board */
124         this.id = this.board.setId(this, 'L');
125         this.board.renderer.drawLine(this);
126         this.board.finalizeAdding(this);
127 
128         this.elType = 'line';
129 
130         /* Add arrow as child to defining points */
131         this.point1.addChild(this);
132         this.point2.addChild(this);
133 
134         this.inherits.push(this.point1, this.point2);
135 
136         this.updateStdform(); // This is needed in the following situation:
137         // * the line is defined by three coordinates
138         // * and it will have a glider
139         // * and board.suspendUpdate() has been called.
140 
141         // create Label
142         this.createLabel();
143 
144         this.methodMap = JXG.deepCopy(this.methodMap, {
145             point1: 'point1',
146             point2: 'point2',
147             getSlope: 'getSlope',
148             getRise: 'getRise',
149             getYIntersect: 'getRise',
150             getAngle: 'getAngle',
151             L: 'L',
152             length: 'L',
153             addTicks: 'addTicks',
154             removeTicks: 'removeTicks',
155             removeAllTicks: 'removeAllTicks'
156         });
157     };
158 
159     JXG.Line.prototype = new GeometryElement();
160 
161     JXG.extend(JXG.Line.prototype, /** @lends JXG.Line.prototype */ {
162         /**
163          * Checks whether (x,y) is near the line.
164          * @param {Number} x Coordinate in x direction, screen coordinates.
165          * @param {Number} y Coordinate in y direction, screen coordinates.
166          * @returns {Boolean} True if (x,y) is near the line, False otherwise.
167          */
168         hasPoint: function (x, y) {
169             // Compute the stdform of the line in screen coordinates.
170             var c = [], s,
171                 v = [1, x, y],
172                 vnew,
173                 p1c, p2c, d, pos, i,
174                 prec,
175                 sw = Type.evaluate(this.visProp.strokewidth);
176 
177             prec = this.board.options.precision.hasPoint + sw * 0.5;
178 
179             c[0] = this.stdform[0] -
180                 this.stdform[1] * this.board.origin.scrCoords[1] / this.board.unitX +
181                 this.stdform[2] * this.board.origin.scrCoords[2] / this.board.unitY;
182             c[1] = this.stdform[1] / this.board.unitX;
183             c[2] = this.stdform[2] / (-this.board.unitY);
184 
185             s = Geometry.distPointLine(v, c);
186             if (isNaN(s) || s > prec) {
187                 return false;
188             }
189 
190             if (Type.evaluate(this.visProp.straightfirs) &&
191                     Type.evaluate(this.visProp.straightlast)) {
192                 return true;
193             }
194 
195             // If the line is a ray or segment we have to check if the projected point is between P1 and P2.
196             p1c = this.point1.coords;
197             p2c = this.point2.coords;
198 
199             // Project the point orthogonally onto the line
200             vnew = [0, c[1], c[2]];
201             // Orthogonal line to c through v
202             vnew = Mat.crossProduct(vnew, v);
203             // Intersect orthogonal line with line
204             vnew = Mat.crossProduct(vnew, c);
205 
206             // Normalize the projected point
207             vnew[1] /= vnew[0];
208             vnew[2] /= vnew[0];
209             vnew[0] = 1;
210 
211             vnew = (new Coords(Const.COORDS_BY_SCREEN, vnew.slice(1), this.board)).usrCoords;
212             d = p1c.distance(Const.COORDS_BY_USER, p2c);
213             p1c = p1c.usrCoords.slice(0);
214             p2c = p2c.usrCoords.slice(0);
215 
216             // The defining points are identical
217             if (d < Mat.eps) {
218                 pos = 0;
219             } else {
220                 /*
221                  * Handle the cases, where one of the defining points is an ideal point.
222                  * d is set to something close to infinity, namely 1/eps.
223                  * The ideal point is (temporarily) replaced by a finite point which has
224                  * distance d from the other point.
225                  * This is accomplishrd by extracting the x- and y-coordinates (x,y)=:v of the ideal point.
226                  * v determines the direction of the line. v is normalized, i.e. set to length 1 by deividing through its length.
227                  * Finally, the new point is the sum of the other point and v*d.
228                  *
229                  */
230 
231                 // At least one point is an ideal point
232                 if (d === Number.POSITIVE_INFINITY) {
233                     d = 1 / Mat.eps;
234 
235                     // The second point is an ideal point
236                     if (Math.abs(p2c[0]) < Mat.eps) {
237                         d /= Geometry.distance([0, 0, 0], p2c);
238                         p2c = [1, p1c[1] + p2c[1] * d, p1c[2] + p2c[2] * d];
239                     // The first point is an ideal point
240                     } else {
241                         d /= Geometry.distance([0, 0, 0], p1c);
242                         p1c = [1, p2c[1] + p1c[1] * d, p2c[2] + p1c[2] * d];
243                     }
244                 }
245                 i = 1;
246                 d = p2c[i] - p1c[i];
247 
248                 if (Math.abs(d) < Mat.eps) {
249                     i = 2;
250                     d = p2c[i] - p1c[i];
251                 }
252                 pos = (vnew[i] - p1c[i]) / d;
253             }
254 
255             if (!Type.evaluate(this.visProp.straightfirst) && pos < 0) {
256                 return false;
257             }
258 
259             return !(!Type.evaluate(this.visProp.straightlast) && pos > 1);
260 
261         },
262 
263         // documented in base/element
264         update: function () {
265             var funps;
266 
267             if (!this.needsUpdate) {
268                 return this;
269             }
270 
271             if (this.constrained) {
272                 if (Type.isFunction(this.funps)) {
273                     funps = this.funps();
274                     if (funps && funps.length && funps.length === 2) {
275                         this.point1 = funps[0];
276                         this.point2 = funps[1];
277                     }
278                 } else {
279                     if (Type.isFunction(this.funp1)) {
280                         funps = this.funp1();
281                         if (Type.isPoint(funps)) {
282                             this.point1 = funps;
283                         } else if (funps && funps.length && funps.length === 2) {
284                             this.point1.setPositionDirectly(Const.COORDS_BY_USER, funps);
285                         }
286                     }
287 
288                     if (Type.isFunction(this.funp2)) {
289                         funps = this.funp2();
290                         if (Type.isPoint(funps)) {
291                             this.point2 = funps;
292                         } else if (funps && funps.length && funps.length === 2) {
293                             this.point2.setPositionDirectly(Const.COORDS_BY_USER, funps);
294                         }
295                     }
296                 }
297             }
298 
299             this.updateSegmentFixedLength();
300             this.updateStdform();
301 
302             if (Type.evaluate(this.visProp.trace)) {
303                 this.cloneToBackground(true);
304             }
305 
306             return this;
307         },
308 
309         /**
310          * Update segments with fixed length and at least one movable point.
311          * @private
312          */
313         updateSegmentFixedLength: function () {
314             var d, dnew, d1, d2, drag1, drag2, x, y;
315 
316             if (!this.hasFixedLength) {
317                 return this;
318             }
319 
320             // Compute the actual length of the segment
321             d = this.point1.Dist(this.point2);
322             // Determine the length the segment ought to have
323             dnew = this.fixedLength();
324             // Distances between the two points and their respective
325             // position before the update
326             d1 = this.fixedLengthOldCoords[0].distance(Const.COORDS_BY_USER, this.point1.coords);
327             d2 = this.fixedLengthOldCoords[1].distance(Const.COORDS_BY_USER, this.point2.coords);
328 
329             // If the position of the points or the fixed length function has been changed we have to work.
330             if (d1 > Mat.eps || d2 > Mat.eps || d !== dnew) {
331                 drag1 = this.point1.isDraggable &&
332                             (this.point1.type !== Const.OBJECT_TYPE_GLIDER) &&
333                             !Type.evaluate(this.point1.visProp.fixed);
334                 drag2 = this.point2.isDraggable &&
335                             (this.point2.type !== Const.OBJECT_TYPE_GLIDER) &&
336                             !Type.evaluate(this.point2.visProp.fixed);
337 
338                 // First case: the two points are different
339                 // Then we try to adapt the point that was not dragged
340                 // If this point can not be moved (e.g. because it is a glider)
341                 // we try move the other point
342                 if (d > Mat.eps) {
343                     if ((d1 > d2 && drag2) ||
344                             (d1 <= d2 && drag2 && !drag1)) {
345                         this.point2.setPositionDirectly(Const.COORDS_BY_USER, [
346                             this.point1.X() + (this.point2.X() - this.point1.X()) * dnew / d,
347                             this.point1.Y() + (this.point2.Y() - this.point1.Y()) * dnew / d
348                         ]);
349                         this.point2.fullUpdate();
350                     } else if ((d1 <= d2 && drag1) ||
351                             (d1 > d2 && drag1 && !drag2)) {
352                         this.point1.setPositionDirectly(Const.COORDS_BY_USER, [
353                             this.point2.X() + (this.point1.X() - this.point2.X()) * dnew / d,
354                             this.point2.Y() + (this.point1.Y() - this.point2.Y()) * dnew / d
355                         ]);
356                         this.point1.fullUpdate();
357                     }
358                     // Second case: the two points are identical. In this situation
359                     // we choose a random direction.
360                 } else {
361                     x = Math.random() - 0.5;
362                     y = Math.random() - 0.5;
363                     d = Math.sqrt(x * x + y * y);
364 
365                     if (drag2) {
366                         this.point2.setPositionDirectly(Const.COORDS_BY_USER, [
367                             this.point1.X() + x * dnew / d,
368                             this.point1.Y() + y * dnew / d
369                         ]);
370                         this.point2.fullUpdate();
371                     } else if (drag1) {
372                         this.point1.setPositionDirectly(Const.COORDS_BY_USER, [
373                             this.point2.X() + x * dnew / d,
374                             this.point2.Y() + y * dnew / d
375                         ]);
376                         this.point1.fullUpdate();
377                     }
378                 }
379                 // Finally, we save the position of the two points.
380                 this.fixedLengthOldCoords[0].setCoordinates(Const.COORDS_BY_USER, this.point1.coords.usrCoords);
381                 this.fixedLengthOldCoords[1].setCoordinates(Const.COORDS_BY_USER, this.point2.coords.usrCoords);
382             }
383             return this;
384         },
385 
386         /**
387          * Updates the stdform derived from the parent point positions.
388          * @private
389          */
390         updateStdform: function () {
391             var v = Mat.crossProduct(this.point1.coords.usrCoords, this.point2.coords.usrCoords);
392 
393             this.stdform[0] = v[0];
394             this.stdform[1] = v[1];
395             this.stdform[2] = v[2];
396             this.stdform[3] = 0;
397 
398             this.normalize();
399         },
400 
401         /**
402          * Uses the boards renderer to update the line.
403          * @private
404          */
405         updateRenderer: function () {
406             var wasReal;
407 
408             if (!this.needsUpdate) {
409                 return this;
410             }
411 
412             if (this.visPropCalc.visible) {
413                 wasReal = this.isReal;
414                 this.isReal = (!isNaN(this.point1.coords.usrCoords[1] + this.point1.coords.usrCoords[2] +
415                         this.point2.coords.usrCoords[1] + this.point2.coords.usrCoords[2]) &&
416                         (Mat.innerProduct(this.stdform, this.stdform, 3) >= Mat.eps * Mat.eps));
417 
418                 if (wasReal && !this.isReal) {
419                     this.updateVisibility(false);
420                 }
421             }
422 
423             if (this.visPropCalc.visible) {
424                 this.board.renderer.updateLine(this);
425             }
426 
427             /* Update the label if visible. */
428             if (this.hasLabel && this.visPropCalc.visible && this.label &&
429                 this.label.visPropCalc.visible && this.isReal) {
430 
431                 this.label.update();
432                 this.board.renderer.updateText(this.label);
433             }
434 
435             // Update rendNode display
436             this.setDisplayRendNode();
437             // if (this.visPropCalc.visible !== this.visPropOld.visible) {
438             //     this.setDisplayRendNode(this.visPropCalc.visible);
439             //     if (this.hasLabel) {
440             //         this.board.renderer.display(this.label, this.label.visPropCalc.visible);
441             //     }
442             // }
443 
444             this.needsUpdate = false;
445             return this;
446         },
447 
448         /**
449          * Used to generate a polynomial for a point p that lies on this line, i.e. p is collinear to
450          * {@link JXG.Line#point1} and {@link JXG.Line#point2}.
451          *
452          * @param {JXG.Point} p The point for that the polynomial is generated.
453          * @returns {Array} An array containing the generated polynomial.
454          * @private
455          */
456         generatePolynomial: function (p) {
457             var u1 = this.point1.symbolic.x,
458                 u2 = this.point1.symbolic.y,
459                 v1 = this.point2.symbolic.x,
460                 v2 = this.point2.symbolic.y,
461                 w1 = p.symbolic.x,
462                 w2 = p.symbolic.y;
463 
464             /*
465              * The polynomial in this case is determined by three points being collinear:
466              *
467              *      U (u1,u2)      W (w1,w2)                V (v1,v2)
468              *  ----x--------------x------------------------x----------------
469              *
470              *  The collinearity condition is
471              *
472              *      u2-w2       w2-v2
473              *     -------  =  -------           (1)
474              *      u1-w1       w1-v1
475              *
476              * Multiplying (1) with denominators and simplifying is
477              *
478              *    u2w1 - u2v1 + w2v1 - u1w2 + u1v2 - w1v2 = 0
479              */
480 
481             return [['(', u2, ')*(', w1, ')-(', u2, ')*(', v1, ')+(', w2, ')*(', v1, ')-(', u1, ')*(', w2, ')+(', u1, ')*(', v2, ')-(', w1, ')*(', v2, ')'].join('')];
482         },
483 
484         /**
485          * Calculates the y intersect of the line.
486          * @returns {Number} The y intersect.
487          */
488         getRise: function () {
489             if (Math.abs(this.stdform[2]) >= Mat.eps) {
490                 return -this.stdform[0] / this.stdform[2];
491             }
492 
493             return Infinity;
494         },
495 
496         /**
497          * Calculates the slope of the line.
498          * @returns {Number} The slope of the line or Infinity if the line is parallel to the y-axis.
499          */
500         getSlope: function () {
501             if (Math.abs(this.stdform[2]) >= Mat.eps) {
502                 return -this.stdform[1] / this.stdform[2];
503             }
504 
505             return Infinity;
506         },
507 
508         /**
509          * Determines the angle between the positive x axis and the line.
510          * @returns {Number}
511          */
512         getAngle: function () {
513             return Math.atan2(-this.stdform[1], this.stdform[2]);
514         },
515 
516         /**
517          * Determines whether the line is drawn beyond {@link JXG.Line#point1} and
518          * {@link JXG.Line#point2} and updates the line.
519          * @param {Boolean} straightFirst True if the Line shall be drawn beyond
520          * {@link JXG.Line#point1}, false otherwise.
521          * @param {Boolean} straightLast True if the Line shall be drawn beyond
522          * {@link JXG.Line#point2}, false otherwise.
523          * @see #straightFirst
524          * @see #straightLast
525          * @private
526          */
527         setStraight: function (straightFirst, straightLast) {
528             this.visProp.straightfirst = straightFirst;
529             this.visProp.straightlast = straightLast;
530 
531             this.board.renderer.updateLine(this);
532             return this;
533         },
534 
535         // documented in geometry element
536         getTextAnchor: function () {
537             return new Coords(Const.COORDS_BY_USER, [0.5 * (this.point2.X() + this.point1.X()), 0.5 * (this.point2.Y() + this.point1.Y())], this.board);
538         },
539 
540         /**
541          * Adjusts Label coords relative to Anchor. DESCRIPTION
542          * @private
543          */
544         setLabelRelativeCoords: function (relCoords) {
545             if (Type.exists(this.label)) {
546                 this.label.relativeCoords = new Coords(Const.COORDS_BY_SCREEN, [relCoords[0], -relCoords[1]], this.board);
547             }
548         },
549 
550         // documented in geometry element
551         getLabelAnchor: function () {
552             var x, y,
553                 fs = 0,
554                 c1 = new Coords(Const.COORDS_BY_USER, this.point1.coords.usrCoords, this.board),
555                 c2 = new Coords(Const.COORDS_BY_USER, this.point2.coords.usrCoords, this.board),
556                 ev_sf = Type.evaluate(this.visProp.straightfirst),
557                 ev_sl = Type.evaluate(this.visProp.straightlast);
558 
559             if (ev_sf || ev_sl) {
560                 Geometry.calcStraight(this, c1, c2, 0);
561             }
562 
563             c1 = c1.scrCoords;
564             c2 = c2.scrCoords;
565 
566             if (!Type.exists(this.label)) {
567                 return new Coords(Const.COORDS_BY_SCREEN, [NaN, NaN], this.board);
568             }
569 
570             switch (Type.evaluate(this.label.visProp.position)) {
571             case 'lft':
572             case 'llft':
573             case 'ulft':
574                 if (c1[1] <= c2[1]) {
575                     x = c1[1];
576                     y = c1[2];
577                 } else {
578                     x = c2[1];
579                     y = c2[2];
580                 }
581                 break;
582             case 'rt':
583             case 'lrt':
584             case 'urt':
585                 if (c1[1] > c2[1]) {
586                     x = c1[1];
587                     y = c1[2];
588                 } else {
589                     x = c2[1];
590                     y = c2[2];
591                 }
592                 break;
593             default:
594                 x = 0.5 * (c1[1] + c2[1]);
595                 y = 0.5 * (c1[2] + c2[2]);
596             }
597 
598             // Correct coordinates if the label seems to be outside of canvas.
599             if (ev_sf || ev_sl) {
600                 if (Type.exists(this.label)) {  // Does not exist during createLabel
601                     fs = Type.evaluate(this.label.visProp.fontsize);
602                 }
603 
604                 if (Math.abs(x) < Mat.eps) {
605                     x = 0;
606                 } else if (this.board.canvasWidth + Mat.eps > x &&
607                             x > this.board.canvasWidth - fs - Mat.eps) {
608                     x = this.board.canvasWidth - fs;
609                 }
610 
611                 if (Mat.eps + fs > y && y > -Mat.eps) {
612                     y = fs;
613                 } else if (this.board.canvasHeight + Mat.eps > y &&
614                             y > this.board.canvasHeight - fs - Mat.eps) {
615                     y = this.board.canvasHeight;
616                 }
617             }
618 
619             return new Coords(Const.COORDS_BY_SCREEN, [x, y], this.board);
620         },
621 
622         // documented in geometry element
623         cloneToBackground: function () {
624             var copy = {}, r, s, er;
625 
626             copy.id = this.id + 'T' + this.numTraces;
627             copy.elementClass = Const.OBJECT_CLASS_LINE;
628             this.numTraces++;
629             copy.point1 = this.point1;
630             copy.point2 = this.point2;
631 
632             copy.stdform = this.stdform;
633 
634             copy.board = this.board;
635 
636             copy.visProp = Type.deepCopy(this.visProp, this.visProp.traceattributes, true);
637             copy.visProp.layer = this.board.options.layer.trace;
638             Type.clearVisPropOld(copy);
639 
640             s = this.getSlope();
641             r = this.getRise();
642             copy.getSlope = function () {
643                 return s;
644             };
645             copy.getRise = function () {
646                 return r;
647             };
648 
649             er = this.board.renderer.enhancedRendering;
650             this.board.renderer.enhancedRendering = true;
651             this.board.renderer.drawLine(copy);
652             this.board.renderer.enhancedRendering = er;
653             this.traces[copy.id] = copy.rendNode;
654 
655             return this;
656         },
657 
658         /**
659          * Add transformations to this line.
660          * @param {JXG.Transformation|Array} transform Either one {@link JXG.Transformation} or an array of
661          * {@link JXG.Transformation}s.
662          * @returns {JXG.Line} Reference to this line object.
663          */
664         addTransform: function (transform) {
665             var i,
666                 list = Type.isArray(transform) ? transform : [transform],
667                 len = list.length;
668 
669             for (i = 0; i < len; i++) {
670                 this.point1.transformations.push(list[i]);
671                 this.point2.transformations.push(list[i]);
672             }
673 
674             return this;
675         },
676 
677         // see GeometryElement.js
678         snapToGrid: function (pos) {
679             var c1, c2, dc, t, ticks,
680                 x, y, sX, sY;
681 
682             if (Type.evaluate(this.visProp.snaptogrid)) {
683                 if (this.parents.length < 3) {    // Line through two points
684                     this.point1.handleSnapToGrid(true, true);
685                     this.point2.handleSnapToGrid(true, true);
686             } else if (Type.exists(pos)) {       // Free line
687                     sX = Type.evaluate(this.visProp.snapsizex);
688                     sY = Type.evaluate(this.visProp.snapsizey);
689 
690                     c1 = new Coords(Const.COORDS_BY_SCREEN, [pos.Xprev, pos.Yprev], this.board);
691 
692                     x = c1.usrCoords[1];
693                     y = c1.usrCoords[2];
694 
695                     if (sX <= 0 && this.board.defaultAxes && this.board.defaultAxes.x.defaultTicks) {
696                         ticks = this.board.defaultAxes.x.defaultTicks;
697                         sX = ticks.ticksDelta * (Type.evaluate(ticks.visProp.minorticks) + 1);
698                     }
699                     if (sY <= 0 && this.board.defaultAxes && this.board.defaultAxes.y.defaultTicks) {
700                         ticks = this.board.defaultAxes.y.defaultTicks;
701                         sY = ticks.ticksDelta * (Type.evaluate(ticks.visProp.minorticks) + 1);
702                     }
703 
704                     // if no valid snap sizes are available, don't change the coords.
705                     if (sX > 0 && sY > 0) {
706                         // projectCoordsToLine
707                         /*
708                         v = [0, this.stdform[1], this.stdform[2]];
709                         v = Mat.crossProduct(v, c1.usrCoords);
710                         c2 = Geometry.meetLineLine(v, this.stdform, 0, this.board);
711                         */
712                         c2 = Geometry.projectPointToLine({coords: c1}, this, this.board);
713 
714                         dc = Statistics.subtract([1, Math.round(x / sX) * sX, Math.round(y / sY) * sY], c2.usrCoords);
715                         t = this.board.create('transform', dc.slice(1), {type: 'translate'});
716                         t.applyOnce([this.point1, this.point2]);
717                     }
718                 }
719             } else {
720                 this.point1.handleSnapToGrid(false, true);
721                 this.point2.handleSnapToGrid(false, true);
722             }
723 
724             return this;
725         },
726 
727         // see element.js
728         snapToPoints: function () {
729             var forceIt = Type.evaluate(this.visProp.snaptopoints);
730 
731             if (this.parents.length < 3) {    // Line through two points
732                 this.point1.handleSnapToPoints(forceIt);
733                 this.point2.handleSnapToPoints(forceIt);
734             }
735 
736             return this;
737         },
738 
739         /**
740          * Treat the line as parametric curve in homogeneous coordinates, where the parameter t runs from 0 to 1.
741          * First we transform the interval [0,1] to [-1,1].
742          * If the line has homogeneous coordinates [c,a,b] = stdform[] then the direction of the line is [b,-a].
743          * Now, we take one finite point that defines the line, i.e. we take either point1 or point2 (in case the line is not the ideal line).
744          * Let the coordinates of that point be [z, x, y].
745          * Then, the curve runs linearly from
746          * [0, b, -a] (t=-1) to [z, x, y] (t=0)
747          * and
748          * [z, x, y] (t=0) to [0, -b, a] (t=1)
749          *
750          * @param {Number} t Parameter running from 0 to 1.
751          * @returns {Number} X(t) x-coordinate of the line treated as parametric curve.
752          * */
753         X: function (t) {
754             var x,
755                 b = this.stdform[2];
756 
757             x = (Math.abs(this.point1.coords.usrCoords[0]) > Mat.eps) ?
758                     this.point1.coords.usrCoords[1] :
759                     this.point2.coords.usrCoords[1];
760 
761             t = (t - 0.5) * 2;
762 
763             return (1 - Math.abs(t)) * x - t * b;
764         },
765 
766         /**
767          * Treat the line as parametric curve in homogeneous coordinates.
768          * See {@link JXG.Line#X} for a detailed description.
769          * @param {Number} t Parameter running from 0 to 1.
770          * @returns {Number} Y(t) y-coordinate of the line treated as parametric curve.
771          */
772         Y: function (t) {
773             var y,
774                 a = this.stdform[1];
775 
776             y = (Math.abs(this.point1.coords.usrCoords[0]) > Mat.eps) ?
777                     this.point1.coords.usrCoords[2] :
778                     this.point2.coords.usrCoords[2];
779 
780             t = (t - 0.5) * 2;
781 
782             return (1 - Math.abs(t)) * y + t * a;
783         },
784 
785         /**
786          * Treat the line as parametric curve in homogeneous coordinates.
787          * See {@link JXG.Line#X} for a detailed description.
788          *
789          * @param {Number} t Parameter running from 0 to 1.
790          * @returns {Number} Z(t) z-coordinate of the line treated as parametric curve.
791          */
792         Z: function (t) {
793             var z = (Math.abs(this.point1.coords.usrCoords[0]) > Mat.eps) ?
794                     this.point1.coords.usrCoords[0] :
795                     this.point2.coords.usrCoords[0];
796 
797             t = (t - 0.5) * 2;
798 
799             return (1 - Math.abs(t)) * z;
800         },
801 
802 
803         /**
804          * The distance between the two points defining the line.
805          * @returns {Number}
806          */
807         L: function () {
808             return this.point1.Dist(this.point2);
809         },
810 
811         /**
812          * Treat the element  as a parametric curve
813          * @private
814          */
815         minX: function () {
816             return 0.0;
817         },
818 
819         /**
820          * Treat the element as parametric curve
821          * @private
822          */
823         maxX: function () {
824             return 1.0;
825         },
826 
827         // documented in geometry element
828         bounds: function () {
829             var p1c = this.point1.coords.usrCoords,
830                 p2c = this.point2.coords.usrCoords;
831 
832             return [Math.min(p1c[1], p2c[1]), Math.max(p1c[2], p2c[2]), Math.max(p1c[1], p2c[1]), Math.min(p1c[2], p2c[2])];
833         },
834 
835         /**
836          * Adds ticks to this line. Ticks can be added to any kind of line: line, arrow, and axis.
837          * @param {JXG.Ticks} ticks Reference to a ticks object which is describing the ticks (color, distance, how many, etc.).
838          * @returns {String} Id of the ticks object.
839          */
840         addTicks: function (ticks) {
841             if (ticks.id === '' || !Type.exists(ticks.id)) {
842                 ticks.id = this.id + '_ticks_' + (this.ticks.length + 1);
843             }
844 
845             this.board.renderer.drawTicks(ticks);
846             this.ticks.push(ticks);
847 
848             return ticks.id;
849         },
850 
851         // documented in GeometryElement.js
852         remove: function () {
853             this.removeAllTicks();
854             GeometryElement.prototype.remove.call(this);
855         },
856 
857         /**
858          * Removes all ticks from a line.
859          */
860         removeAllTicks: function () {
861             var t;
862 
863             for (t = this.ticks.length; t > 0; t--) {
864                 this.removeTicks(this.ticks[t - 1]);
865             }
866 
867             this.ticks = [];
868             this.board.update();
869         },
870 
871         /**
872          * Removes ticks identified by parameter named tick from this line.
873          * @param {JXG.Ticks} tick Reference to tick object to remove.
874          */
875         removeTicks: function (tick) {
876             var t, j;
877 
878             if (Type.exists(this.defaultTicks) && this.defaultTicks === tick) {
879                 this.defaultTicks = null;
880             }
881 
882             for (t = this.ticks.length; t > 0; t--) {
883                 if (this.ticks[t - 1] === tick) {
884                     this.board.removeObject(this.ticks[t - 1]);
885 
886                     if (this.ticks[t - 1].ticks) {
887                         for (j = 0; j < this.ticks[t - 1].ticks.length; j++) {
888                             if (Type.exists(this.ticks[t - 1].labels[j])) {
889                                 this.board.removeObject(this.ticks[t - 1].labels[j]);
890                             }
891                         }
892                     }
893 
894                     delete this.ticks[t - 1];
895                     break;
896                 }
897             }
898         },
899 
900         // hideElement: function () {
901         //     var i;
902         //
903         //     GeometryElement.prototype.hideElement.call(this);
904         //
905         //     for (i = 0; i < this.ticks.length; i++) {
906         //         this.ticks[i].hideElement();
907         //     }
908         // },
909         //
910         // showElement: function () {
911         //     var i;
912         //     GeometryElement.prototype.showElement.call(this);
913         //
914         //     for (i = 0; i < this.ticks.length; i++) {
915         //         this.ticks[i].showElement();
916         //     }
917         // }
918     });
919 
920     /**
921      * @class This element is used to provide a constructor for a general line. A general line is given by two points. By setting additional properties
922      * a line can be used as an arrow and/or axis.
923      * @pseudo
924      * @description
925      * @name Line
926      * @augments JXG.Line
927      * @constructor
928      * @type JXG.Line
929      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
930      * @param {JXG.Point,array,function_JXG.Point,array,function} point1,point2 Parent elements can be two elements either of type {@link JXG.Point} or array of
931      * numbers describing the coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
932      * It is possible to provide a function returning an array or a point, instead of providing an array or a point.
933      * @param {Number,function_Number,function_Number,function} c,a,b A line can also be created providing three numbers. The line is then described by
934      * the set of solutions of the equation <tt>a*x+b*y+c*z = 0</tt>. It is possible to provide three functions returning numbers, too.
935      * @param {function} f This function must return an array containing three numbers forming the line's homogeneous coordinates.
936      * @example
937      * // Create a line using point and coordinates/
938      * // The second point will be fixed and invisible.
939      * var p1 = board.create('point', [4.5, 2.0]);
940      * var l1 = board.create('line', [p1, [1.0, 1.0]]);
941      * </pre><div class="jxgbox" id="c0ae3461-10c4-4d39-b9be-81d74759d122" style="width: 300px; height: 300px;"></div>
942      * <script type="text/javascript">
943      *   var glex1_board = JXG.JSXGraph.initBoard('c0ae3461-10c4-4d39-b9be-81d74759d122', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
944      *   var glex1_p1 = glex1_board.create('point', [4.5, 2.0]);
945      *   var glex1_l1 = glex1_board.create('line', [glex1_p1, [1.0, 1.0]]);
946      * </script><pre>
947      * @example
948      * // Create a line using three coordinates
949      * var l1 = board.create('line', [1.0, -2.0, 3.0]);
950      * </pre><div class="jxgbox" id="cf45e462-f964-4ba4-be3a-c9db94e2593f" style="width: 300px; height: 300px;"></div>
951      * <script type="text/javascript">
952      *   var glex2_board = JXG.JSXGraph.initBoard('cf45e462-f964-4ba4-be3a-c9db94e2593f', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
953      *   var glex2_l1 = glex2_board.create('line', [1.0, -2.0, 3.0]);
954      * </script><pre>
955      */
956     JXG.createLine = function (board, parents, attributes) {
957         var ps, el, p1, p2, i, attr,
958             c = [],
959             constrained = false,
960             isDraggable;
961 
962         /**
963          * The line is defined by two points or coordinates of two points.
964          * In the latter case, the points are created.
965          */
966         if (parents.length === 2) {
967             // point 1 given by coordinates
968             if (Type.isArray(parents[0]) && parents[0].length > 1) {
969                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
970                 p1 = board.create('point', parents[0], attr);
971             } else if (Type.isString(parents[0]) || Type.isPoint(parents[0])) {
972                 p1 =  board.select(parents[0]);
973             } else if (Type.isFunction(parents[0]) && Type.isPoint(parents[0]())) {
974                 p1 = parents[0]();
975                 constrained = true;
976             } else if (Type.isFunction(parents[0]) && parents[0]().length && parents[0]().length >= 2) {
977                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
978                 p1 = Point.createPoint(board, parents[0](), attr);
979                 constrained = true;
980             } else {
981                 throw new Error("JSXGraph: Can't create line with parent types '" +
982                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
983                     "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
984             }
985 
986             // point 2 given by coordinates
987             if (Type.isArray(parents[1]) && parents[1].length > 1) {
988                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
989                 p2 = board.create('point', parents[1], attr);
990             } else if (Type.isString(parents[1]) || Type.isPoint(parents[1])) {
991                 p2 =  board.select(parents[1]);
992             } else if (Type.isFunction(parents[1]) &&  Type.isPoint(parents[1]()) ) {
993                 p2 = parents[1]();
994                 constrained = true;
995             } else if (Type.isFunction(parents[1]) && parents[1]().length && parents[1]().length >= 2) {
996                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
997                 p2 = Point.createPoint(board, parents[1](), attr);
998                 constrained = true;
999             } else {
1000                 throw new Error("JSXGraph: Can't create line with parent types '" +
1001                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1002                     "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1003             }
1004 
1005             attr = Type.copyAttributes(attributes, board.options, 'line');
1006 
1007             el = new JXG.Line(board, p1, p2, attr);
1008             if (constrained) {
1009                 el.constrained = true;
1010                 el.funp1 = parents[0];
1011                 el.funp2 = parents[1];
1012             } else {
1013                 el.isDraggable = true;
1014             }
1015 
1016             //if (!el.constrained) {
1017             el.setParents([p1.id, p2.id]);
1018             //}
1019 
1020          // Line is defined by three homogeneous coordinates.
1021          // Also in this case points are created.
1022         } else if (parents.length === 3) {
1023             // free line
1024             isDraggable = true;
1025             for (i = 0; i < 3; i++) {
1026                 if (Type.isNumber(parents[i])) {
1027                     // createFunction will just wrap a function around our constant number
1028                     // that does nothing else but to return that number.
1029                     c[i] = Type.createFunction(parents[i]);
1030                 } else if (Type.isFunction(parents[i])) {
1031                     c[i] = parents[i];
1032                     isDraggable = false;
1033                 } else {
1034                     throw new Error("JSXGraph: Can't create line with parent types '" +
1035                         (typeof parents[0]) + "' and '" + (typeof parents[1]) + "' and '" + (typeof parents[2]) + "'." +
1036                         "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1037                 }
1038             }
1039 
1040             // point 1 is the midpoint between (0,c,-b) and point 2. => point1 is finite.
1041             attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
1042             if (isDraggable) {
1043                 p1 = board.create('point', [
1044                     c[2]() * c[2]() + c[1]() * c[1](),
1045                     c[2]() - c[1]() * c[0]() + c[2](),
1046                     -c[1]() - c[2]() * c[0]() - c[1]()
1047                 ], attr);
1048             } else {
1049                 p1 = board.create('point', [
1050                     function () {
1051                         return (c[2]() * c[2]() + c[1]() * c[1]()) * 0.5;
1052                     },
1053                     function () {
1054                         return (c[2]() - c[1]() * c[0]() + c[2]()) * 0.5;
1055                     },
1056                     function () {
1057                         return (-c[1]() - c[2]() * c[0]() - c[1]()) * 0.5;
1058                     }], attr);
1059             }
1060 
1061             // point 2: (b^2+c^2,-ba+c,-ca-b)
1062             attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1063             if (isDraggable) {
1064                 p2 = board.create('point', [
1065                     c[2]() * c[2]() + c[1]() * c[1](),
1066                     -c[1]() * c[0]() + c[2](),
1067                     -c[2]() * c[0]() - c[1]()
1068                 ], attr);
1069             } else {
1070                 p2 = board.create('point', [
1071                     function () {
1072                         return c[2]() * c[2]() + c[1]() * c[1]();
1073                     },
1074                     function () {
1075                         return -c[1]() * c[0]() + c[2]();
1076                     },
1077                     function () {
1078                         return -c[2]() * c[0]() - c[1]();
1079                     }], attr);
1080             }
1081 
1082             // If the line will have a glider and board.suspendUpdate() has been called, we
1083             // need to compute the initial position of the two points p1 and p2.
1084             p1.prepareUpdate().update();
1085             p2.prepareUpdate().update();
1086             attr = Type.copyAttributes(attributes, board.options, 'line');
1087             el = new JXG.Line(board, p1, p2, attr);
1088             // Not yet working, because the points are not draggable.
1089             el.isDraggable = isDraggable;
1090             el.setParents([p1, p2]);
1091 
1092         // The parent array contains a function which returns two points.
1093         } else if (parents.length === 1 && Type.isFunction(parents[0]) && parents[0]().length === 2 &&
1094                 Type.isPoint(parents[0]()[0]) &&
1095                 Type.isPoint(parents[0]()[1])) {
1096             ps = parents[0]();
1097             attr = Type.copyAttributes(attributes, board.options, 'line');
1098             el = new JXG.Line(board, ps[0], ps[1], attr);
1099             el.constrained = true;
1100             el.funps = parents[0];
1101             el.setParents(ps);
1102 
1103         } else if (parents.length === 1 && Type.isFunction(parents[0]) && parents[0]().length === 3 &&
1104                 Type.isNumber(parents[0]()[0]) &&
1105                 Type.isNumber(parents[0]()[1]) &&
1106                 Type.isNumber(parents[0]()[2])) {
1107             ps = parents[0];
1108 
1109             attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
1110             p1 = board.create('point', [
1111                 function () {
1112                     var c = ps();
1113 
1114                     return [
1115                         (c[2] * c[2] + c[1] * c[1]) * 0.5,
1116                         (c[2] - c[1] * c[0] + c[2]) * 0.5,
1117                         (-c[1] - c[2] * c[0] - c[1]) * 0.5
1118                     ];
1119                 }], attr);
1120 
1121             attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1122             p2 = board.create('point', [
1123                 function () {
1124                     var c = ps();
1125 
1126                     return [
1127                         c[2] * c[2] + c[1] * c[1],
1128                         -c[1] * c[0] + c[2],
1129                         -c[2] * c[0] - c[1]
1130                     ];
1131                 }], attr);
1132 
1133             attr = Type.copyAttributes(attributes, board.options, 'line');
1134             el = new JXG.Line(board, p1, p2, attr);
1135 
1136             el.constrained = true;
1137             el.funps = parents[0];
1138             el.setParents([p1, p2]);
1139 
1140         } else {
1141             throw new Error("JSXGraph: Can't create line with parent types '" +
1142                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1143                 "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1144         }
1145 
1146         return el;
1147     };
1148 
1149     JXG.registerElement('line', JXG.createLine);
1150 
1151     /**
1152      * @class This element is used to provide a constructor for a segment.
1153      * It's strictly spoken just a wrapper for element {@link Line} with {@link JXG.Line#straightFirst}
1154      * and {@link JXG.Line#straightLast} properties set to false. If there is a third variable then the
1155      * segment has a fixed length (which may be a function, too).
1156      * @pseudo
1157      * @description
1158      * @name Segment
1159      * @augments JXG.Line
1160      * @constructor
1161      * @type JXG.Line
1162      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1163      * @param {JXG.Point,array_JXG.Point,array} point1,point2 Parent elements can be two elements either of type {@link JXG.Point}
1164      * or array of numbers describing the
1165      * coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
1166      * @param {number,function} length (optional) The points are adapted - if possible - such that their distance
1167      * has a this value.
1168      * @see Line
1169      * @example
1170      * // Create a segment providing two points.
1171      *   var p1 = board.create('point', [4.5, 2.0]);
1172      *   var p2 = board.create('point', [1.0, 1.0]);
1173      *   var l1 = board.create('segment', [p1, p2]);
1174      * </pre><div class="jxgbox" id="d70e6aac-7c93-4525-a94c-a1820fa38e2f" style="width: 300px; height: 300px;"></div>
1175      * <script type="text/javascript">
1176      *   var slex1_board = JXG.JSXGraph.initBoard('d70e6aac-7c93-4525-a94c-a1820fa38e2f', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1177      *   var slex1_p1 = slex1_board.create('point', [4.5, 2.0]);
1178      *   var slex1_p2 = slex1_board.create('point', [1.0, 1.0]);
1179      *   var slex1_l1 = slex1_board.create('segment', [slex1_p1, slex1_p2]);
1180      * </script><pre>
1181      *
1182      * @example
1183      * // Create a segment providing two points.
1184      *   var p1 = board.create('point', [4.0, 1.0]);
1185      *   var p2 = board.create('point', [1.0, 1.0]);
1186      *   var l1 = board.create('segment', [p1, p2]);
1187      *   var p3 = board.create('point', [4.0, 2.0]);
1188      *   var p4 = board.create('point', [1.0, 2.0]);
1189      *   var l2 = board.create('segment', [p3, p4, 3]);
1190      *   var p5 = board.create('point', [4.0, 3.0]);
1191      *   var p6 = board.create('point', [1.0, 4.0]);
1192      *   var l3 = board.create('segment', [p5, p6, function(){ return l1.L();} ]);
1193      * </pre><div class="jxgbox" id="617336ba-0705-4b2b-a236-c87c28ef25be" style="width: 300px; height: 300px;"></div>
1194      * <script type="text/javascript">
1195      *   var slex2_board = JXG.JSXGraph.initBoard('617336ba-0705-4b2b-a236-c87c28ef25be', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1196      *   var slex2_p1 = slex2_board.create('point', [4.0, 1.0]);
1197      *   var slex2_p2 = slex2_board.create('point', [1.0, 1.0]);
1198      *   var slex2_l1 = slex2_board.create('segment', [slex2_p1, slex2_p2]);
1199      *   var slex2_p3 = slex2_board.create('point', [4.0, 2.0]);
1200      *   var slex2_p4 = slex2_board.create('point', [1.0, 2.0]);
1201      *   var slex2_l2 = slex2_board.create('segment', [slex2_p3, slex2_p4, 3]);
1202      *   var slex2_p5 = slex2_board.create('point', [4.0, 2.0]);
1203      *   var slex2_p6 = slex2_board.create('point', [1.0, 2.0]);
1204      *   var slex2_l3 = slex2_board.create('segment', [slex2_p5, slex2_p6, function(){ return slex2_l1.L();}]);
1205      * </script><pre>
1206      *
1207      */
1208     JXG.createSegment = function (board, parents, attributes) {
1209         var el, attr;
1210 
1211         attributes.straightFirst = false;
1212         attributes.straightLast = false;
1213         attr = Type.copyAttributes(attributes, board.options, 'segment');
1214 
1215         el = board.create('line', parents.slice(0, 2), attr);
1216 
1217         if (parents.length === 3) {
1218             el.hasFixedLength = true;
1219 
1220             if (Type.isNumber(parents[2])) {
1221                 el.fixedLength = function () {
1222                     return parents[2];
1223                 };
1224             } else if (Type.isFunction(parents[2])) {
1225                 el.fixedLength = parents[2];
1226             } else {
1227                 throw new Error("JSXGraph: Can't create segment with third parent type '" +
1228                     (typeof parents[2]) + "'." +
1229                     "\nPossible third parent types: number or function");
1230             }
1231 
1232             el.getParents = function() {
1233                 return this.parents.concat(this.fixedLength());
1234             };
1235 
1236             el.fixedLengthOldCoords = [];
1237             el.fixedLengthOldCoords[0] = new Coords(Const.COORDS_BY_USER, el.point1.coords.usrCoords.slice(1, 3), board);
1238             el.fixedLengthOldCoords[1] = new Coords(Const.COORDS_BY_USER, el.point2.coords.usrCoords.slice(1, 3), board);
1239         }
1240 
1241         el.elType = 'segment';
1242 
1243         return el;
1244     };
1245 
1246     JXG.registerElement('segment', JXG.createSegment);
1247 
1248     /**
1249      * @class This element is used to provide a constructor for arrow, which is just a wrapper for element {@link Line} with {@link JXG.Line#straightFirst}
1250      * and {@link JXG.Line#straightLast} properties set to false and {@link JXG.Line#lastArrow} set to true.
1251      * @pseudo
1252      * @description
1253      * @name Arrow
1254      * @augments JXG.Line
1255      * @constructor
1256      * @type JXG.Line
1257      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1258      * @param {JXG.Point,array_JXG.Point,array} point1,point2 Parent elements can be two elements either of type {@link JXG.Point} or array of numbers describing the
1259      * coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
1260      * @param {Number_Number_Number} a,b,c A line can also be created providing three numbers. The line is then described by the set of solutions
1261      * of the equation <tt>a*x+b*y+c*z = 0</tt>.
1262      * @see Line
1263      * @example
1264      * // Create an arrow providing two points.
1265      *   var p1 = board.create('point', [4.5, 2.0]);
1266      *   var p2 = board.create('point', [1.0, 1.0]);
1267      *   var l1 = board.create('arrow', [p1, p2]);
1268      * </pre><div class="jxgbox" id="1d26bd22-7d6d-4018-b164-4c8bc8d22ccf" style="width: 300px; height: 300px;"></div>
1269      * <script type="text/javascript">
1270      *   var alex1_board = JXG.JSXGraph.initBoard('1d26bd22-7d6d-4018-b164-4c8bc8d22ccf', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1271      *   var alex1_p1 = alex1_board.create('point', [4.5, 2.0]);
1272      *   var alex1_p2 = alex1_board.create('point', [1.0, 1.0]);
1273      *   var alex1_l1 = alex1_board.create('arrow', [alex1_p1, alex1_p2]);
1274      * </script><pre>
1275      */
1276     JXG.createArrow = function (board, parents, attributes) {
1277         var el;
1278 
1279         attributes.firstArrow = false;
1280         attributes.lastArrow = true;
1281         el = board.create('line', parents, attributes).setStraight(false, false);
1282         //el.setArrow(false, true);
1283         el.type = Const.OBJECT_TYPE_VECTOR;
1284         el.elType = 'arrow';
1285 
1286         return el;
1287     };
1288 
1289     JXG.registerElement('arrow', JXG.createArrow);
1290 
1291     /**
1292      * @class This element is used to provide a constructor for an axis. It's strictly spoken just a wrapper for element {@link Line} with {@link JXG.Line#straightFirst}
1293      * and {@link JXG.Line#straightLast} properties set to true. Additionally {@link JXG.Line#lastArrow} is set to true and default {@link Ticks} will be created.
1294      * @pseudo
1295      * @description
1296      * @name Axis
1297      * @augments JXG.Line
1298      * @constructor
1299      * @type JXG.Line
1300      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1301      * @param {JXG.Point,array_JXG.Point,array} point1,point2 Parent elements can be two elements either of type {@link JXG.Point} or array of numbers describing the
1302      * coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
1303      * @param {Number_Number_Number} a,b,c A line can also be created providing three numbers. The line is then described by the set of solutions
1304      * of the equation <tt>a*x+b*y+c*z = 0</tt>.
1305      * @example
1306      * // Create an axis providing two coord pairs.
1307      *   var l1 = board.create('axis', [[0.0, 1.0], [1.0, 1.3]]);
1308      * </pre><div class="jxgbox" id="4f414733-624c-42e4-855c-11f5530383ae" style="width: 300px; height: 300px;"></div>
1309      * <script type="text/javascript">
1310      *   var axex1_board = JXG.JSXGraph.initBoard('4f414733-624c-42e4-855c-11f5530383ae', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1311      *   var axex1_l1 = axex1_board.create('axis', [[0.0, 1.0], [1.0, 1.3]]);
1312      * </script><pre>
1313      */
1314     JXG.createAxis = function (board, parents, attributes) {
1315         var attr, attr_ticks, el, els, dist;
1316 
1317         // Arrays oder Punkte, mehr brauchen wir nicht.
1318         if ((Type.isArray(parents[0]) || Type.isPoint(parents[0])) && (Type.isArray(parents[1]) || Type.isPoint(parents[1]))) {
1319             attr = Type.copyAttributes(attributes, board.options, 'axis');
1320             el = board.create('line', parents, attr);
1321             el.type = Const.OBJECT_TYPE_AXIS;
1322             el.isDraggable = false;
1323             el.point1.isDraggable = false;
1324             el.point2.isDraggable = false;
1325 
1326             for (els in el.ancestors) {
1327                 if (el.ancestors.hasOwnProperty(els)) {
1328                     el.ancestors[els].type = Const.OBJECT_TYPE_AXISPOINT;
1329                 }
1330             }
1331 
1332             attr_ticks = Type.copyAttributes(attributes, board.options, 'axis', 'ticks');
1333             if (Type.exists(attr_ticks.ticksdistance)) {
1334                 dist = attr_ticks.ticksdistance;
1335             } else if (Type.isArray(attr_ticks.ticks)) {
1336                 dist = attr_ticks.ticks;
1337             } else {
1338                 dist = 1.0;
1339             }
1340 
1341             /**
1342              * The ticks attached to the axis.
1343              * @memberOf Axis.prototype
1344              * @name defaultTicks
1345              * @type JXG.Ticks
1346              */
1347             el.defaultTicks = board.create('ticks', [el, dist], attr_ticks);
1348             el.defaultTicks.dump = false;
1349             el.elType = 'axis';
1350             el.subs = {
1351                 ticks: el.defaultTicks
1352             };
1353             el.inherits.push(el.defaultTicks);
1354 
1355         } else {
1356             throw new Error("JSXGraph: Can't create axis with parent types '" +
1357                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1358                 "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]]");
1359         }
1360 
1361         return el;
1362     };
1363 
1364     JXG.registerElement('axis', JXG.createAxis);
1365 
1366     /**
1367      * @class With the element tangent the slope of a line, circle, or curve in a certain point can be visualized. A tangent is always constructed
1368      * by a glider on a line, circle, or curve and describes the tangent in the glider point on that line, circle, or curve.
1369      * @pseudo
1370      * @description
1371      * @name Tangent
1372      * @augments JXG.Line
1373      * @constructor
1374      * @type JXG.Line
1375      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1376      * @param {Glider} g A glider on a line, circle, or curve.
1377      * @example
1378      * // Create a tangent providing a glider on a function graph
1379      *   var c1 = board.create('curve', [function(t){return t},function(t){return t*t*t;}]);
1380      *   var g1 = board.create('glider', [0.6, 1.2, c1]);
1381      *   var t1 = board.create('tangent', [g1]);
1382      * </pre><div class="jxgbox" id="7b7233a0-f363-47dd-9df5-4018d0d17a98" style="width: 400px; height: 400px;"></div>
1383      * <script type="text/javascript">
1384      *   var tlex1_board = JXG.JSXGraph.initBoard('7b7233a0-f363-47dd-9df5-4018d0d17a98', {boundingbox: [-6, 6, 6, -6], axis: true, showcopyright: false, shownavigation: false});
1385      *   var tlex1_c1 = tlex1_board.create('curve', [function(t){return t},function(t){return t*t*t;}]);
1386      *   var tlex1_g1 = tlex1_board.create('glider', [0.6, 1.2, tlex1_c1]);
1387      *   var tlex1_t1 = tlex1_board.create('tangent', [tlex1_g1]);
1388      * </script><pre>
1389      */
1390     JXG.createTangent = function (board, parents, attributes) {
1391         var p, c, g, f, j, el, tangent;
1392 
1393         // One arguments: glider on line, circle or curve
1394         if (parents.length === 1) {
1395             p = parents[0];
1396             c = p.slideObject;
1397         // Two arguments: (point,F"|conic) or (line|curve|circle|conic,point). // Not yet: curve!
1398         } else if (parents.length === 2) {
1399             // In fact, for circles and conics it is the polar
1400             if (Type.isPoint(parents[0])) {
1401                 p = parents[0];
1402                 c = parents[1];
1403             } else if (Type.isPoint(parents[1])) {
1404                 c = parents[0];
1405                 p = parents[1];
1406             } else {
1407                 throw new Error("JSXGraph: Can't create tangent with parent types '" +
1408                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1409                     "\nPossible parent types: [glider], [point,line|curve|circle|conic]");
1410             }
1411         } else {
1412             throw new Error("JSXGraph: Can't create tangent with parent types '" +
1413                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1414                 "\nPossible parent types: [glider], [point,line|curve|circle|conic]");
1415         }
1416 
1417         if (c.elementClass === Const.OBJECT_CLASS_LINE) {
1418             tangent = board.create('line', [c.point1, c.point2], attributes);
1419             tangent.glider = p;
1420         } else if (c.elementClass === Const.OBJECT_CLASS_CURVE && c.type !== Const.OBJECT_TYPE_CONIC) {
1421             if (Type.evaluate(c.visProp.curvetype) !== 'plot') {
1422                 g = c.X;
1423                 f = c.Y;
1424                 tangent = board.create('line', [
1425                     function () {
1426                         return -p.X() * Numerics.D(f)(p.position) + p.Y() * Numerics.D(g)(p.position);
1427                     },
1428                     function () {
1429                         return Numerics.D(f)(p.position);
1430                     },
1431                     function () {
1432                         return -Numerics.D(g)(p.position);
1433                     }
1434                 ], attributes);
1435                 p.addChild(tangent);
1436 
1437                 // this is required for the geogebra reader to display a slope
1438                 tangent.glider = p;
1439             } else {  // curveType 'plot'
1440                 // equation of the line segment: 0 = y*(x1-x2) + x*(y2-y1) + y1*x2-x1*y2
1441                 tangent = board.create('line', [
1442                     function () {
1443                         var i = Math.floor(p.position);
1444 
1445                         if (i === c.numberPoints - 1) {
1446                             i--;
1447                         }
1448 
1449                         if (i < 0) {
1450                             return 1;
1451                         }
1452 
1453                         return c.Y(i) * c.X(i + 1) - c.X(i) * c.Y(i + 1);
1454                     },
1455                     function () {
1456                         var i = Math.floor(p.position);
1457 
1458                         if (i === c.numberPoints - 1) {
1459                             i--;
1460                         }
1461 
1462                         if (i < 0) {
1463                             return 0;
1464                         }
1465 
1466                         return c.Y(i + 1) - c.Y(i);
1467                     },
1468                     function () {
1469                         var i = Math.floor(p.position);
1470 
1471                         if (i === c.numberPoints - 1) {
1472                             i--;
1473                         }
1474 
1475                         if (i < 0) {
1476                             return 0.0;
1477                         }
1478 
1479                         return c.X(i) - c.X(i + 1);
1480                     }], attributes);
1481 
1482                 p.addChild(tangent);
1483 
1484                 // this is required for the geogebra reader to display a slope
1485                 tangent.glider = p;
1486             }
1487         } else if (c.type === Const.OBJECT_TYPE_TURTLE) {
1488             tangent = board.create('line', [
1489                 function () {
1490                     var i = Math.floor(p.position);
1491 
1492                     // run through all curves of this turtle
1493                     for (j = 0; j < c.objects.length; j++) {
1494                         el = c.objects[j];
1495 
1496                         if (el.type === Const.OBJECT_TYPE_CURVE) {
1497                             if (i < el.numberPoints) {
1498                                 break;
1499                             }
1500 
1501                             i -= el.numberPoints;
1502                         }
1503                     }
1504 
1505                     if (i === el.numberPoints - 1) {
1506                         i--;
1507                     }
1508 
1509                     if (i < 0) {
1510                         return 1;
1511                     }
1512 
1513                     return el.Y(i) * el.X(i + 1) - el.X(i) * el.Y(i + 1);
1514                 },
1515                 function () {
1516                     var i = Math.floor(p.position);
1517 
1518                     // run through all curves of this turtle
1519                     for (j = 0; j < c.objects.length; j++) {
1520                         el = c.objects[j];
1521 
1522                         if (el.type === Const.OBJECT_TYPE_CURVE) {
1523                             if (i < el.numberPoints) {
1524                                 break;
1525                             }
1526 
1527                             i -= el.numberPoints;
1528                         }
1529                     }
1530 
1531                     if (i === el.numberPoints - 1) {
1532                         i--;
1533                     }
1534                     if (i < 0) {
1535                         return 0;
1536                     }
1537 
1538                     return el.Y(i + 1) - el.Y(i);
1539                 },
1540                 function () {
1541                     var i = Math.floor(p.position);
1542 
1543                     // run through all curves of this turtle
1544                     for (j = 0; j < c.objects.length; j++) {
1545                         el = c.objects[j];
1546                         if (el.type === Const.OBJECT_TYPE_CURVE) {
1547                             if (i < el.numberPoints) {
1548                                 break;
1549                             }
1550                             i -= el.numberPoints;
1551                         }
1552                     }
1553                     if (i === el.numberPoints - 1) {
1554                         i--;
1555                     }
1556 
1557                     if (i < 0) {
1558                         return 0;
1559                     }
1560 
1561                     return el.X(i) - el.X(i + 1);
1562                 }], attributes);
1563             p.addChild(tangent);
1564 
1565             // this is required for the geogebra reader to display a slope
1566             tangent.glider = p;
1567         } else if (c.elementClass === Const.OBJECT_CLASS_CIRCLE || c.type === Const.OBJECT_TYPE_CONIC) {
1568             // If p is not on c, the tangent is the polar.
1569             // This construction should work on conics, too. p has to lie on c.
1570             tangent = board.create('line', [
1571                 function () {
1572                     return Mat.matVecMult(c.quadraticform, p.coords.usrCoords)[0];
1573                 },
1574                 function () {
1575                     return Mat.matVecMult(c.quadraticform, p.coords.usrCoords)[1];
1576                 },
1577                 function () {
1578                     return Mat.matVecMult(c.quadraticform, p.coords.usrCoords)[2];
1579                 }], attributes);
1580 
1581             p.addChild(tangent);
1582             // this is required for the geogebra reader to display a slope
1583             tangent.glider = p;
1584         }
1585 
1586         if (!Type.exists(tangent)) {
1587             throw new Error('JSXGraph: Couldn\'t create tangent with the given parents.');
1588         }
1589 
1590         tangent.elType = 'tangent';
1591         tangent.type = Const.OBJECT_TYPE_TANGENT;
1592         tangent.setParents(parents);
1593 
1594         return tangent;
1595     };
1596 
1597     /**
1598      * @class This element is used to provide a constructor for the radical axis with respect to two circles with distinct centers.
1599      * The angular bisector of the polar lines of the circle centers with respect to the other circle is always the radical axis.
1600      * The radical axis passes through the intersection points when the circles intersect.
1601      * When a circle about the midpoint of circle centers, passing through the circle centers, intersects the circles, the polar lines pass through those intersection points.
1602      * @pseudo
1603      * @description
1604      * @name RadicalAxis
1605      * @augments JXG.Line
1606      * @constructor
1607      * @type JXG.Line
1608      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1609      * @param {JXG.Circle} circle Circle one of the two respective circles.
1610      * @param {JXG.Circle} circle Circle the other of the two respective circles.
1611      * @example
1612      * // Create the radical axis line with respect to two circles
1613      *   var board = JXG.JSXGraph.initBoard('7b7233a0-f363-47dd-9df5-5018d0d17a98', {boundingbox: [-1, 9, 9, -1], axis: true, showcopyright: false, shownavigation: false});
1614      *   var p1 = board.create('point', [2, 3]);
1615      *   var p2 = board.create('point', [1, 4]);
1616      *   var c1 = board.create('circle', [p1, p2]);
1617      *   var p3 = board.create('point', [6, 5]);
1618      *   var p4 = board.create('point', [8, 6]);
1619      *   var c2 = board.create('circle', [p3, p4]);
1620      *   var r1 = board.create('radicalaxis', [c1, c2]);
1621      * </pre><div class="jxgbox" id='7b7233a0-f363-47dd-9df5-5018d0d17a98' class='jxgbox' style='width:400px; height:400px;'></div>
1622      * <script type='text/javascript'>
1623      *   var rlex1_board = JXG.JSXGraph.initBoard('7b7233a0-f363-47dd-9df5-5018d0d17a98', {boundingbox: [-1, 9, 9, -1], axis: true, showcopyright: false, shownavigation: false});
1624      *   var rlex1_p1 = rlex1_board.create('point', [2, 3]);
1625      *   var rlex1_p2 = rlex1_board.create('point', [1, 4]);
1626      *   var rlex1_c1 = rlex1_board.create('circle', [rlex1_p1, rlex1_p2]);
1627      *   var rlex1_p3 = rlex1_board.create('point', [6, 5]);
1628      *   var rlex1_p4 = rlex1_board.create('point', [8, 6]);
1629      *   var rlex1_c2 = rlex1_board.create('circle', [rlex1_p3, rlex1_p4]);
1630      *   var rlex1_r1 = rlex1_board.create('radicalaxis', [rlex1_c1, rlex1_c2]);
1631      * </script><pre>
1632      */
1633     JXG.createRadicalAxis = function (board, parents, attributes) {
1634         var el, el1, el2;
1635 
1636         if (parents.length !== 2 ||
1637                 parents[0].elementClass !== Const.OBJECT_CLASS_CIRCLE ||
1638                 parents[1].elementClass !== Const.OBJECT_CLASS_CIRCLE) {
1639             // Failure
1640             throw new Error("JSXGraph: Can't create 'radical axis' with parent types '" +
1641                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1642                 "\nPossible parent type: [circle,circle]");
1643         }
1644 
1645         el1 = board.select(parents[0]);
1646         el2 = board.select(parents[1]);
1647 
1648         el = board.create('line', [function () {
1649             var a = el1.stdform,
1650                 b = el2.stdform;
1651 
1652             return Mat.matVecMult(Mat.transpose([a.slice(0, 3), b.slice(0, 3)]), [b[3], -a[3]]);
1653         }], attributes);
1654 
1655         el.elType = 'radicalaxis';
1656         el.setParents([el1.id, el2.id]);
1657 
1658         el1.addChild(el);
1659         el2.addChild(el);
1660 
1661         return el;
1662     };
1663 
1664     /**
1665      * @class This element is used to provide a constructor for the polar line of a point with respect to a conic or a circle.
1666      * @pseudo
1667      * @description The polar line is the unique reciprocal relationship of a point with respect to a conic.
1668      * The lines through the intersections of a conic and the polar line of a point with respect to that conic and through that point are tangent to the conic.
1669      * A point on a conic has the polar line of that point with respect to that conic as the tangent line to that conic at that point.
1670      * See {@link http://en.wikipedia.org/wiki/Pole_and_polar} for more information on pole and polar.
1671      * @name PolarLine
1672      * @augments JXG.Line
1673      * @constructor
1674      * @type JXG.Line
1675      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1676      * @param {JXG.Conic,JXG.Circle_JXG.Point} el1,el2 or
1677      * @param {JXG.Point_JXG.Conic,JXG.Circle} el1,el2 The result will be the polar line of the point with respect to the conic or the circle.
1678      * @example
1679      * // Create the polar line of a point with respect to a conic
1680      * var p1 = board.create('point', [-1, 2]);
1681      * var p2 = board.create('point', [ 1, 4]);
1682      * var p3 = board.create('point', [-1,-2]);
1683      * var p4 = board.create('point', [ 0, 0]);
1684      * var p5 = board.create('point', [ 4,-2]);
1685      * var c1 = board.create('conic',[p1,p2,p3,p4,p5]);
1686      * var p6 = board.create('point', [-1, 1]);
1687      * var l1 = board.create('polarline', [c1, p6]);
1688      * </pre><div class="jxgbox" id='7b7233a0-f363-47dd-9df5-6018d0d17a98' class='jxgbox' style='width:400px; height:400px;'></div>
1689      * <script type='text/javascript'>
1690      * var plex1_board = JXG.JSXGraph.initBoard('7b7233a0-f363-47dd-9df5-6018d0d17a98', {boundingbox: [-3, 5, 5, -3], axis: true, showcopyright: false, shownavigation: false});
1691      * var plex1_p1 = plex1_board.create('point', [-1, 2]);
1692      * var plex1_p2 = plex1_board.create('point', [ 1, 4]);
1693      * var plex1_p3 = plex1_board.create('point', [-1,-2]);
1694      * var plex1_p4 = plex1_board.create('point', [ 0, 0]);
1695      * var plex1_p5 = plex1_board.create('point', [ 4,-2]);
1696      * var plex1_c1 = plex1_board.create('conic',[plex1_p1,plex1_p2,plex1_p3,plex1_p4,plex1_p5]);
1697      * var plex1_p6 = plex1_board.create('point', [-1, 1]);
1698      * var plex1_l1 = plex1_board.create('polarline', [plex1_c1, plex1_p6]);
1699      * </script><pre>
1700      * @example
1701      * // Create the polar line of a point with respect to a circle.
1702      * var p1 = board.create('point', [ 1, 1]);
1703      * var p2 = board.create('point', [ 2, 3]);
1704      * var c1 = board.create('circle',[p1,p2]);
1705      * var p3 = board.create('point', [ 6, 6]);
1706      * var l1 = board.create('polarline', [c1, p3]);
1707      * </pre><div class="jxgbox" id='7b7233a0-f363-47dd-9df5-7018d0d17a98' class='jxgbox' style='width:400px; height:400px;'></div>
1708      * <script type='text/javascript'>
1709      * var plex2_board = JXG.JSXGraph.initBoard('7b7233a0-f363-47dd-9df5-7018d0d17a98', {boundingbox: [-3, 7, 7, -3], axis: true, showcopyright: false, shownavigation: false});
1710      * var plex2_p1 = plex2_board.create('point', [ 1, 1]);
1711      * var plex2_p2 = plex2_board.create('point', [ 2, 3]);
1712      * var plex2_c1 = plex2_board.create('circle',[plex2_p1,plex2_p2]);
1713      * var plex2_p3 = plex2_board.create('point', [ 6, 6]);
1714      * var plex2_l1 = plex2_board.create('polarline', [plex2_c1, plex2_p3]);
1715      * </script><pre>
1716      */
1717     JXG.createPolarLine = function (board, parents, attributes) {
1718         var el, el1, el2,
1719             firstParentIsConic, secondParentIsConic,
1720             firstParentIsPoint, secondParentIsPoint;
1721 
1722         if (parents.length > 1) {
1723             firstParentIsConic = (parents[0].type === Const.OBJECT_TYPE_CONIC ||
1724                 parents[0].elementClass === Const.OBJECT_CLASS_CIRCLE);
1725             secondParentIsConic = (parents[1].type === Const.OBJECT_TYPE_CONIC ||
1726                 parents[1].elementClass === Const.OBJECT_CLASS_CIRCLE);
1727 
1728             firstParentIsPoint = (Type.isPoint(parents[0]));
1729             secondParentIsPoint = (Type.isPoint(parents[1]));
1730         }
1731 
1732         if (parents.length !== 2 ||
1733                 !((firstParentIsConic && secondParentIsPoint) ||
1734                     (firstParentIsPoint && secondParentIsConic))) {
1735             // Failure
1736             throw new Error("JSXGraph: Can't create 'polar line' with parent types '" +
1737                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1738                 "\nPossible parent type: [conic|circle,point], [point,conic|circle]");
1739         }
1740 
1741         if (secondParentIsPoint) {
1742             el1 = board.select(parents[0]);
1743             el2 = board.select(parents[1]);
1744         } else {
1745             el1 = board.select(parents[1]);
1746             el2 = board.select(parents[0]);
1747         }
1748 
1749         // Polar lines have been already provided in the tangent element.
1750         el = board.create('tangent', [el1, el2], attributes);
1751 
1752         el.elType = 'polarline';
1753         return el;
1754     };
1755 
1756     /**
1757      * Register the element type tangent at JSXGraph
1758      * @private
1759      */
1760     JXG.registerElement('tangent', JXG.createTangent);
1761     JXG.registerElement('polar', JXG.createTangent);
1762     JXG.registerElement('radicalaxis', JXG.createRadicalAxis);
1763     JXG.registerElement('polarline', JXG.createPolarLine);
1764 
1765     return {
1766         Line: JXG.Line,
1767         createLine: JXG.createLine,
1768         createTangent: JXG.createTangent,
1769         createPolar: JXG.createTangent,
1770         createSegment: JXG.createSegment,
1771         createAxis: JXG.createAxis,
1772         createArrow: JXG.createArrow,
1773         createRadicalAxis: JXG.createRadicalAxis,
1774         createPolarLine: JXG.createPolarLine
1775     };
1776 });
1777