1 /*
  2     Copyright 2008-2022
  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, console: true, window: true*/
 34 /*jslint nomen: true, plusplus: true*/
 35 
 36 /* depends:
 37  jxg
 38  options
 39  math/math
 40  math/geometry
 41  math/numerics
 42  base/coords
 43  base/constants
 44  base/element
 45  parser/geonext
 46  utils/type
 47   elements:
 48    transform
 49  */
 50 
 51 /**
 52  * @fileoverview The geometry object Point is defined in this file. Point stores all
 53  * style and functional properties that are required to draw and move a point on
 54  * a board.
 55  */
 56 
 57 define([
 58     'jxg', 'options', 'math/math', 'math/geometry', 'base/constants', 'base/element',
 59     'utils/type', 'base/coordselement'
 60 ], function (JXG, Options, Mat, Geometry, Const, GeometryElement, Type, CoordsElement) {
 61 
 62     "use strict";
 63 
 64     /**
 65      * A point is the basic geometric element. Based on points lines and circles can be constructed which can be intersected
 66      * which in turn are points again which can be used to construct new lines, circles, polygons, etc. This class holds methods for
 67      * all kind of points like free points, gliders, and intersection points.
 68      * @class Creates a new point object. Do not use this constructor to create a point. Use {@link JXG.Board#create} with
 69      * type {@link Point}, {@link Glider}, or {@link Intersection} instead.
 70      * @augments JXG.GeometryElement
 71      * @augments JXG.CoordsElement
 72      * @param {string|JXG.Board} board The board the new point is drawn on.
 73      * @param {Array} coordinates An array with the user coordinates of the point.
 74      * @param {Object} attributes An object containing visual properties like in {@link JXG.Options#point} and
 75      * {@link JXG.Options#elements}, and optional a name and an id.
 76      * @see JXG.Board#generateName
 77      */
 78     JXG.Point = function (board, coordinates, attributes) {
 79         this.constructor(board, attributes, Const.OBJECT_TYPE_POINT, Const.OBJECT_CLASS_POINT);
 80         this.element = this.board.select(attributes.anchor);
 81         this.coordsConstructor(coordinates);
 82 
 83         this.elType = 'point';
 84 
 85         /* Register point at board. */
 86         this.id = this.board.setId(this, 'P');
 87         this.board.renderer.drawPoint(this);
 88         this.board.finalizeAdding(this);
 89 
 90         this.createLabel();
 91 
 92     };
 93 
 94     /**
 95      * Inherits here from {@link JXG.GeometryElement}.
 96      */
 97     JXG.Point.prototype = new GeometryElement();
 98     Type.copyPrototypeMethods(JXG.Point, CoordsElement, 'coordsConstructor');
 99 
100     JXG.extend(JXG.Point.prototype, /** @lends JXG.Point.prototype */ {
101         /**
102          * Checks whether (x,y) is near the point.
103          * @param {Number} x Coordinate in x direction, screen coordinates.
104          * @param {Number} y Coordinate in y direction, screen coordinates.
105          * @returns {Boolean} True if (x,y) is near the point, False otherwise.
106          * @private
107          */
108         hasPoint: function (x, y) {
109             var coordsScr = this.coords.scrCoords, r,
110                 prec, type,
111                 unit = Type.evaluate(this.visProp.sizeunit);
112 
113             if (Type.isObject(Type.evaluate(this.visProp.precision))) {
114                 type = this.board._inputDevice;
115                 prec = Type.evaluate(this.visProp.precision[type]);
116             } else {
117                 // 'inherit'
118                 prec = this.board.options.precision.hasPoint;
119             }
120             r = parseFloat(Type.evaluate(this.visProp.size));
121             if (unit === 'user') {
122                 r *= Math.sqrt(this.board.unitX * this.board.unitY);
123             }
124 
125             r += parseFloat(Type.evaluate(this.visProp.strokewidth)) * 0.5;
126             if (r < prec) {
127                 r = prec;
128             }
129 
130             return ((Math.abs(coordsScr[1] - x) < r + 2) && (Math.abs(coordsScr[2] - y) < r + 2));
131         },
132 
133         /**
134          * Updates the position of the point.
135          */
136         update: function (fromParent) {
137             if (!this.needsUpdate) {
138                 return this;
139             }
140 
141             this.updateCoords(fromParent);
142 
143             if (Type.evaluate(this.visProp.trace)) {
144                 this.cloneToBackground(true);
145             }
146 
147             return this;
148         },
149 
150         /**
151          * Applies the transformations of the element to {@link JXG.Point#baseElement}.
152          * Point transformations are relative to a base element.
153          * @param {Boolean} fromParent True if the drag comes from a child element. This is the case if a line
154          *    through two points is dragged. Otherwise, the element is the drag element and we apply the
155          *    the inverse transformation to the baseElement if is different from the element.
156          * @returns {JXG.CoordsElement} Reference to this object.
157          */
158         updateTransform: function (fromParent) {
159             var c, i;
160 
161             if (this.transformations.length === 0 || this.baseElement === null) {
162                 return this;
163             }
164 
165             if (this === this.baseElement) {
166                 // Case of bindTo
167                 c = this.transformations[0].apply(this.baseElement, 'self');
168                 this.coords.setCoordinates(Const.COORDS_BY_USER, c);
169             } else {
170                 c = this.transformations[0].apply(this.baseElement);
171             }
172             this.coords.setCoordinates(Const.COORDS_BY_USER, c);
173 
174             for (i = 1; i < this.transformations.length; i++) {
175                 this.coords.setCoordinates(Const.COORDS_BY_USER, this.transformations[i].apply(this));
176             }
177             return this;
178         },
179 
180         /**
181          * Calls the renderer to update the drawing.
182          * @private
183          */
184         updateRenderer: function () {
185             this.updateRendererGeneric('updatePoint');
186             return this;
187         },
188 
189         // documented in JXG.GeometryElement
190         bounds: function () {
191             return this.coords.usrCoords.slice(1).concat(this.coords.usrCoords.slice(1));
192         },
193 
194         /**
195          * Convert the point to intersection point and update the construction.
196          * To move the point visual onto the intersection, a call of board update is necessary.
197          *
198          * @param {String|Object} el1, el2, i, j The intersecting objects and the numbers.
199          **/
200         makeIntersection: function (el1, el2, i, j) {
201             var func;
202 
203             el1 = this.board.select(el1);
204             el2 = this.board.select(el2);
205 
206             func = Geometry.intersectionFunction(this.board, el1, el2, i, j,
207                     Type.evaluate(this.visProp.alwaysintersect));
208             this.addConstraint([func]);
209 
210             try {
211                 el1.addChild(this);
212                 el2.addChild(this);
213             } catch (e) {
214                 throw new Error("JSXGraph: Can't create 'intersection' with parent types '" +
215                     (typeof el1) + "' and '" + (typeof el2) + "'.");
216             }
217 
218             this.type = Const.OBJECT_TYPE_INTERSECTION;
219             this.elType = 'intersection';
220             this.parents = [el1.id, el2.id, i, j];
221 
222             this.generatePolynomial = function () {
223                 var poly1 = el1.generatePolynomial(this),
224                     poly2 = el2.generatePolynomial(this);
225 
226                 if ((poly1.length === 0) || (poly2.length === 0)) {
227                     return [];
228                 }
229 
230                 return [poly1[0], poly2[0]];
231             };
232 
233             this.prepareUpdate().update();
234         },
235 
236         /**
237          * Set the style of a point.
238          * Used for GEONExT import and should not be used to set the point's face and size.
239          * @param {Number} i Integer to determine the style.
240          * @private
241          */
242         setStyle: function (i) {
243             var facemap = [
244                 // 0-2
245                 'cross', 'cross', 'cross',
246                 // 3-6
247                 'circle', 'circle', 'circle', 'circle',
248                 // 7-9
249                 'square', 'square', 'square',
250                 // 10-12
251                 'plus', 'plus', 'plus'
252             ], sizemap = [
253                 // 0-2
254                 2, 3, 4,
255                 // 3-6
256                 1, 2, 3, 4,
257                 // 7-9
258                 2, 3, 4,
259                 // 10-12
260                 2, 3, 4
261             ];
262 
263             this.visProp.face = facemap[i];
264             this.visProp.size = sizemap[i];
265 
266             this.board.renderer.changePointStyle(this);
267             return this;
268         },
269 
270         /**
271          * @deprecated Use JXG#normalizePointFace instead
272          * @param s
273          * @returns {*}
274          */
275         normalizeFace: function (s) {
276             JXG.deprecated('Point.normalizeFace()', 'JXG.normalizePointFace()');
277             return Options.normalizePointFace(s);
278         },
279 
280         /**
281          * Set the face of a point element.
282          * @param {String} f String which determines the face of the point. See {@link JXG.GeometryElement#face} for a list of available faces.
283          * @see JXG.GeometryElement#face
284          * @deprecated Use setAttribute()
285          */
286         face: function (f) {
287             JXG.deprecated('Point.face()', 'Point.setAttribute()');
288             this.setAttribute({face: f});
289         },
290 
291         /**
292          * Set the size of a point element
293          * @param {Number} s Integer which determines the size of the point.
294          * @see JXG.GeometryElement#size
295          * @deprecated Use setAttribute()
296          */
297         size: function (s) {
298             JXG.deprecated('Point.size()', 'Point.setAttribute()');
299             this.setAttribute({size: s});
300         },
301 
302         /**
303          * Test if the point is on (is incident with) element "el".
304          *
305          * @param {JXG.GeometryElement} el
306          * @param {Number} tol
307          * @returns {Boolean}
308          *
309          * @example
310          * var circ = board.create('circle', [[-2, -2], 1]);
311          * var seg = board.create('segment', [[-1, -3], [0,0]]);
312          * var line = board.create('line', [[1, 3], [2, -2]]);
313          * var po = board.create('point', [-1, 0], {color: 'blue'});
314          * var curve = board.create('functiongraph', ['sin(x)'], {strokeColor: 'blue'});
315          * var pol = board.create('polygon', [[2,2], [4,2], [4,3]], {strokeColor: 'blue'});
316          *
317          * var point = board.create('point', [-1, 1], {
318          *               attractors: [line, seg, circ, po, curve, pol],
319          *               attractorDistance: 0.2
320          *             });
321          *
322          * var txt = board.create('text', [-4, 3, function() {
323          *              return 'point on line: ' + point.isOn(line) + '<br>' +
324          *                 'point on seg: ' + point.isOn(seg) + '<br>' +
325          *                 'point on circ = ' + point.isOn(circ) + '<br>' +
326          *                 'point on point = ' + point.isOn(po) + '<br>' +
327          *                 'point on curve = ' + point.isOn(curve) + '<br>' +
328          *                 'point on polygon = ' + point.isOn(pol) + '<br>';
329          * }]);
330          *
331          * </pre><div id="JXG6c7d7404-758a-44eb-802c-e9644b9fab71" class="jxgbox" style="width: 300px; height: 300px;"></div>
332          * <script type="text/javascript">
333          *     (function() {
334          *         var board = JXG.JSXGraph.initBoard('JXG6c7d7404-758a-44eb-802c-e9644b9fab71',
335          *             {boundingbox: [-8, 8, 8,-8], axis: true, showcopyright: false, shownavigation: false});
336          *     var circ = board.create('circle', [[-2, -2], 1]);
337          *     var seg = board.create('segment', [[-1, -3], [0,0]]);
338          *     var line = board.create('line', [[1, 3], [2, -2]]);
339          *     var po = board.create('point', [-1, 0], {color: 'blue'});
340          *     var curve = board.create('functiongraph', ['sin(x)'], {strokeColor: 'blue'});
341          *     var pol = board.create('polygon', [[2,2], [4,2], [4,3]], {strokeColor: 'blue'});
342          *
343          *     var point = board.create('point', [-1, 1], {
344          *                   attractors: [line, seg, circ, po, curve, pol],
345          *                   attractorDistance: 0.2
346          *                 });
347          *
348          *     var txt = board.create('text', [-4, 3, function() {
349          *             return 'point on line: ' + point.isOn(line) + '<br>' +
350          *                     'point on seg: ' + point.isOn(seg) + '<br>' +
351          *                     'point on circ = ' + point.isOn(circ) + '<br>' +
352          *                     'point on point = ' + point.isOn(po) + '<br>' +
353          *                     'point on curve = ' + point.isOn(curve) + '<br>' +
354          *                     'point on polygon = ' + point.isOn(pol) + '<br>';
355          *     }]);
356          *
357          *     })();
358          *
359          * </script><pre>
360          *
361          */
362         isOn: function(el, tol) {
363             var arr, crds;
364 
365             tol = tol || Mat.eps;
366 
367             if (Type.isPoint(el)) {
368                 return this.Dist(el) < tol;
369             } else if (el.elementClass === Const.OBJECT_CLASS_LINE) {
370                 if (el.elType === 'segment' && !Type.evaluate(this.visProp.alwaysintersect)) {
371                     arr = JXG.Math.Geometry.projectCoordsToSegment(
372                                 this.coords.usrCoords,
373                                 el.point1.coords.usrCoords,
374                                 el.point2.coords.usrCoords);
375                     if (arr[1] >= 0 && arr[1] <= 1 &&
376                         Geometry.distPointLine(this.coords.usrCoords, el.stdform) < tol) {
377                            return true;
378                     } else {
379                         return false;
380                     }
381                 } else {
382                     return Geometry.distPointLine(this.coords.usrCoords, el.stdform) < tol;
383                 }
384             } else if (el.elementClass === Const.OBJECT_CLASS_CIRCLE) {
385                 if (Type.evaluate(el.visProp.hasinnerpoints)) {
386                     return this.Dist(el.center) < el.Radius() + tol;
387                 }
388                 return Math.abs(this.Dist(el.center) - el.Radius()) < tol;
389             } else if (el.elementClass === Const.OBJECT_CLASS_CURVE) {
390                 crds = Geometry.projectPointToCurve(this, el, this.board)[0];
391                 return Geometry.distance(this.coords.usrCoords, crds.usrCoords, 3) < tol;
392             } else if (el.type === Const.OBJECT_TYPE_POLYGON) {
393                 if (Type.evaluate(el.visProp.hasinnerpoints)) {
394                     if (el.pnpoly(this.coords.usrCoords[1], this.coords.usrCoords[2], JXG.COORDS_BY_USER)) {
395                         return true;
396                     }
397                 }
398                 arr = Geometry.projectCoordsToPolygon(this.coords.usrCoords, el);
399                 return Geometry.distance(this.coords.usrCoords, arr, 3) < tol;
400             } else if (el.type === Const.OBJECT_TYPE_TURTLE) {
401                 crds = Geometry.projectPointToTurtle(this, el, this.board);
402                 return Geometry.distance(this.coords.usrCoords, crds.usrCoords, 3) < tol;
403             }
404 
405             // TODO: Arc, Sector
406             return false;
407         },
408 
409         // Already documented in GeometryElement
410         cloneToBackground: function () {
411             var copy = {};
412 
413             copy.id = this.id + 'T' + this.numTraces;
414             this.numTraces += 1;
415 
416             copy.coords = this.coords;
417             copy.visProp = Type.deepCopy(this.visProp, this.visProp.traceattributes, true);
418             copy.visProp.layer = this.board.options.layer.trace;
419             copy.elementClass = Const.OBJECT_CLASS_POINT;
420             copy.board = this.board;
421             Type.clearVisPropOld(copy);
422 
423             copy.visPropCalc = {
424                 visible: Type.evaluate(copy.visProp.visible)
425             };
426 
427             this.board.renderer.drawPoint(copy);
428             this.traces[copy.id] = copy.rendNode;
429 
430             return this;
431         }
432 
433     });
434 
435     /**
436      * @class This element is used to provide a constructor for a general point. A free point is created if the given parent elements are all numbers
437      * and the property fixed is not set or set to false. If one or more parent elements is not a number but a string containing a GEONE<sub>x</sub>T
438      * constraint or a function the point will be considered as constrained). That means that the user won't be able to change the point's
439      * position directly.
440      * @pseudo
441      * @description
442      * @name Point
443      * @augments JXG.Point
444      * @constructor
445      * @type JXG.Point
446      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
447      * @param {Number,string,function_Number,string,function_Number,string,function} z_,x,y Parent elements can be two or three elements of type number, a string containing a GEONE<sub>x</sub>T
448      * constraint, or a function which takes no parameter and returns a number. Every parent element determines one coordinate. If a coordinate is
449      * given by a number, the number determines the initial position of a free point. If given by a string or a function that coordinate will be constrained
450      * that means the user won't be able to change the point's position directly by mouse because it will be calculated automatically depending on the string
451      * or the function's return value. If two parent elements are given the coordinates will be interpreted as 2D affine Euclidean coordinates, if three such
452      * parent elements are given they will be interpreted as homogeneous coordinates.
453      * @param {JXG.Point_JXG.Transformation_Array} Point,Transformation A point can also be created providing a transformation or an array of transformations.
454      * The resulting point is a clone of the base point transformed by the given Transformation. {@see JXG.Transformation}.
455      *
456      * @example
457      * // Create a free point using affine Euclidean coordinates
458      * var p1 = board.create('point', [3.5, 2.0]);
459      * </pre><div class="jxgbox" id="JXG672f1764-7dfa-4abc-a2c6-81fbbf83e44b" style="width: 200px; height: 200px;"></div>
460      * <script type="text/javascript">
461      *   var board = JXG.JSXGraph.initBoard('JXG672f1764-7dfa-4abc-a2c6-81fbbf83e44b', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
462      *   var p1 = board.create('point', [3.5, 2.0]);
463      * </script><pre>
464      * @example
465      * // Create a constrained point using anonymous function
466      * var p2 = board.create('point', [3.5, function () { return p1.X(); }]);
467      * </pre><div class="jxgbox" id="JXG4fd4410c-3383-4e80-b1bb-961f5eeef224" style="width: 200px; height: 200px;"></div>
468      * <script type="text/javascript">
469      *   var fpex1_board = JXG.JSXGraph.initBoard('JXG4fd4410c-3383-4e80-b1bb-961f5eeef224', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
470      *   var fpex1_p1 = fpex1_board.create('point', [3.5, 2.0]);
471      *   var fpex1_p2 = fpex1_board.create('point', [3.5, function () { return fpex1_p1.X(); }]);
472      * </script><pre>
473      * @example
474      * // Create a point using transformations
475      * var trans = board.create('transform', [2, 0.5], {type:'scale'});
476      * var p3 = board.create('point', [p2, trans]);
477      * </pre><div class="jxgbox" id="JXG630afdf3-0a64-46e0-8a44-f51bd197bb8d" style="width: 400px; height: 400px;"></div>
478      * <script type="text/javascript">
479      *   var fpex2_board = JXG.JSXGraph.initBoard('JXG630afdf3-0a64-46e0-8a44-f51bd197bb8d', {boundingbox: [-1, 9, 9, -1], axis: true, showcopyright: false, shownavigation: false});
480      *   var fpex2_trans = fpex2_board.create('transform', [2, 0.5], {type:'scale'});
481      *   var fpex2_p2 = fpex2_board.create('point', [3.5, 2.0]);
482      *   var fpex2_p3 = fpex2_board.create('point', [fpex2_p2, fpex2_trans]);
483      * </script><pre>
484      */
485     JXG.createPoint = function (board, parents, attributes) {
486         var el, attr;
487 
488         attr = Type.copyAttributes(attributes, board.options, 'point');
489         el = CoordsElement.create(JXG.Point, board, parents, attr);
490         if (!el) {
491             throw new Error("JSXGraph: Can't create point with parent types '" +
492                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
493                     "\nPossible parent types: [x,y], [z,x,y], [element,transformation]");
494         }
495 
496         return el;
497     };
498 
499     /**
500      * @class This element is used to provide a constructor for a glider point.
501      * @pseudo
502      * @description A glider is a point which lives on another geometric element like a line, circle, curve, turtle.
503      * @name Glider
504      * @augments JXG.Point
505      * @constructor
506      * @type JXG.Point
507      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
508      * @param {Number_Number_Number_JXG.GeometryElement} z_,x_,y_,GlideObject Parent elements can be two or three elements of type number and the object the glider lives on.
509      * The coordinates are completely optional. If not given the origin is used. If you provide two numbers for coordinates they will be interpreted as affine Euclidean
510      * coordinates, otherwise they will be interpreted as homogeneous coordinates. In any case the point will be projected on the glide object.
511      * @example
512      * // Create a glider with user defined coordinates. If the coordinates are not on
513      * // the circle (like in this case) the point will be projected onto the circle.
514      * var p1 = board.create('point', [2.0, 2.0]);
515      * var c1 = board.create('circle', [p1, 2.0]);
516      * var p2 = board.create('glider', [2.0, 1.5, c1]);
517      * </pre><div class="jxgbox" id="JXG4f65f32f-e50a-4b50-9b7c-f6ec41652930" style="width: 300px; height: 300px;"></div>
518      * <script type="text/javascript">
519      *   var gpex1_board = JXG.JSXGraph.initBoard('JXG4f65f32f-e50a-4b50-9b7c-f6ec41652930', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
520      *   var gpex1_p1 = gpex1_board.create('point', [2.0, 2.0]);
521      *   var gpex1_c1 = gpex1_board.create('circle', [gpex1_p1, 2.0]);
522      *   var gpex1_p2 = gpex1_board.create('glider', [2.0, 1.5, gpex1_c1]);
523      * </script><pre>
524      * @example
525      * // Create a glider with default coordinates (1,0,0). Same premises as above.
526      * var p1 = board.create('point', [2.0, 2.0]);
527      * var c1 = board.create('circle', [p1, 2.0]);
528      * var p2 = board.create('glider', [c1]);
529      * </pre><div class="jxgbox" id="JXG4de7f181-631a-44b1-a12f-bc4d995609e8" style="width: 200px; height: 200px;"></div>
530      * <script type="text/javascript">
531      *   var gpex2_board = JXG.JSXGraph.initBoard('JXG4de7f181-631a-44b1-a12f-bc4d995609e8', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
532      *   var gpex2_p1 = gpex2_board.create('point', [2.0, 2.0]);
533      *   var gpex2_c1 = gpex2_board.create('circle', [gpex2_p1, 2.0]);
534      *   var gpex2_p2 = gpex2_board.create('glider', [gpex2_c1]);
535      * </script><pre>
536      *@example
537      * //animate example 2
538      * var p1 = board.create('point', [2.0, 2.0]);
539      * var c1 = board.create('circle', [p1, 2.0]);
540      * var p2 = board.create('glider', [c1]);
541      * var button1 = board.create('button', [1, 7, 'start animation',function(){p2.startAnimation(1,4)}]);
542      * var button2 = board.create('button', [1, 5, 'stop animation',function(){p2.stopAnimation()}]);
543      * </pre><div class="jxgbox" id="JXG4de7f181-631a-44b1-a12f-bc4d133709e8" style="width: 200px; height: 200px;"></div>
544      * <script type="text/javascript">
545      *   var gpex3_board = JXG.JSXGraph.initBoard('JXG4de7f181-631a-44b1-a12f-bc4d133709e8', {boundingbox: [-1, 10, 10, -1], axis: true, showcopyright: false, shownavigation: false});
546      *   var gpex3_p1 = gpex3_board.create('point', [2.0, 2.0]);
547      *   var gpex3_c1 = gpex3_board.create('circle', [gpex3_p1, 2.0]);
548      *   var gpex3_p2 = gpex3_board.create('glider', [gpex3_c1]);
549      *   gpex3_board.create('button', [1, 7, 'start animation',function(){gpex3_p2.startAnimation(1,4)}]);
550      *   gpex3_board.create('button', [1, 5, 'stop animation',function(){gpex3_p2.stopAnimation()}]);
551      * </script><pre>
552      */
553     JXG.createGlider = function (board, parents, attributes) {
554         var el, coords,
555             attr = Type.copyAttributes(attributes, board.options, 'glider');
556 
557         if (parents.length === 1) {
558             coords = [0, 0];
559         } else {
560             coords = parents.slice(0, 2);
561         }
562         el = board.create('point', coords, attr);
563 
564         // eltype is set in here
565         el.makeGlider(parents[parents.length - 1]);
566 
567         return el;
568     };
569 
570 
571     /**
572      * @class An intersection point is a point which lives on two JSXGraph elements, i.e. it is one point of the the set
573      * consisting of the intersection points of the two elements. The following element types can be (mutually) intersected: line, circle,
574      * curve, polygon, polygonal chain.
575      *
576      * @pseudo
577      * @name Intersection
578      * @augments JXG.Point
579      * @constructor
580      * @type JXG.Point
581      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
582      * @param {JXG.Line,JXG.Circle_JXG.Line,JXG.Circle_Number} el1,el2,i The result will be a intersection point on el1 and el2. i determines the
583      * intersection point if two points are available: <ul>
584      *   <li>i==0: use the positive square root,</li>
585      *   <li>i==1: use the negative square root.</li></ul>
586      * @example
587      * // Create an intersection point of circle and line
588      * var p1 = board.create('point', [2.0, 2.0]);
589      * var c1 = board.create('circle', [p1, 2.0]);
590      *
591      * var p2 = board.create('point', [2.0, 2.0]);
592      * var p3 = board.create('point', [2.0, 2.0]);
593      * var l1 = board.create('line', [p2, p3]);
594      *
595      * var i = board.create('intersection', [c1, l1, 0]);
596      * </pre><div class="jxgbox" id="JXGe5b0e190-5200-4bc3-b995-b6cc53dc5dc0" style="width: 300px; height: 300px;"></div>
597      * <script type="text/javascript">
598      *   var ipex1_board = JXG.JSXGraph.initBoard('JXGe5b0e190-5200-4bc3-b995-b6cc53dc5dc0', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
599      *   var ipex1_p1 = ipex1_board.create('point', [4.0, 4.0]);
600      *   var ipex1_c1 = ipex1_board.create('circle', [ipex1_p1, 2.0]);
601      *   var ipex1_p2 = ipex1_board.create('point', [1.0, 1.0]);
602      *   var ipex1_p3 = ipex1_board.create('point', [5.0, 3.0]);
603      *   var ipex1_l1 = ipex1_board.create('line', [ipex1_p2, ipex1_p3]);
604      *   var ipex1_i = ipex1_board.create('intersection', [ipex1_c1, ipex1_l1, 0]);
605      * </script><pre>
606      */
607     JXG.createIntersectionPoint = function (board, parents, attributes) {
608         var el, el1, el2, func, i, j,
609             attr = Type.copyAttributes(attributes, board.options, 'intersection');
610 
611         // make sure we definitely have the indices
612         parents.push(0, 0);
613 
614         el1 = board.select(parents[0]);
615         el2 = board.select(parents[1]);
616 
617         i = parents[2] || 0;
618         j = parents[3] || 0;
619 
620         el = board.create('point', [0, 0, 0], attr);
621 
622         // el.visProp.alwaysintersect is evaluated as late as in the returned function
623         func = Geometry.intersectionFunction(board, el1, el2, i, j, el.visProp.alwaysintersect);
624         el.addConstraint([func]);
625 
626         try {
627             el1.addChild(el);
628             el2.addChild(el);
629         } catch (e) {
630             throw new Error("JSXGraph: Can't create 'intersection' with parent types '" +
631                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'.");
632         }
633 
634         el.type = Const.OBJECT_TYPE_INTERSECTION;
635         el.elType = 'intersection';
636         el.setParents([el1.id, el2.id]);
637 
638         /**
639          * Array of length 2 containing the numbers i and j.
640          * The intersection point is i-th intersection point.
641          * j is unused.
642          * @type Array
643          * @private
644          */
645         el.intersectionNumbers = [i, j];
646         el.getParents = function() {
647             return this.parents.concat(this.intersectionNumbers);
648         };
649 
650         el.generatePolynomial = function () {
651             var poly1 = el1.generatePolynomial(el),
652                 poly2 = el2.generatePolynomial(el);
653 
654             if ((poly1.length === 0) || (poly2.length === 0)) {
655                 return [];
656             }
657 
658             return [poly1[0], poly2[0]];
659         };
660 
661         return el;
662     };
663 
664     /**
665      * @class This element is used to provide a constructor for the "other" intersection point.
666      * @pseudo
667      * @description An intersection point is a point which lives on two Lines or Circles or one Line and one Circle at the same time, i.e.
668      * an intersection point of the two elements. Additionally, one intersection point is provided. The function returns the other intersection point.
669      * @name OtherIntersection
670      * @augments JXG.Point
671      * @constructor
672      * @type JXG.Point
673      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
674      * @param {JXG.Line,JXG.Circle_JXG.Line,JXG.Circle_JXG.Point} el1,el2,p The result will be a intersection point on el1 and el2. i determines the
675      * intersection point different from p:
676      * @example
677      * // Create an intersection point of circle and line
678      * var p1 = board.create('point', [2.0, 2.0]);
679      * var c1 = board.create('circle', [p1, 2.0]);
680      *
681      * var p2 = board.create('point', [2.0, 2.0]);
682      * var p3 = board.create('point', [2.0, 2.0]);
683      * var l1 = board.create('line', [p2, p3]);
684      *
685      * var i = board.create('intersection', [c1, l1, 0]);
686      * var j = board.create('otherintersection', [c1, l1, i]);
687      * </pre><div class="jxgbox" id="JXG45e25f12-a1de-4257-a466-27a2ae73614c" style="width: 300px; height: 300px;"></div>
688      * <script type="text/javascript">
689      *   var ipex2_board = JXG.JSXGraph.initBoard('JXG45e25f12-a1de-4257-a466-27a2ae73614c', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
690      *   var ipex2_p1 = ipex2_board.create('point', [4.0, 4.0]);
691      *   var ipex2_c1 = ipex2_board.create('circle', [ipex2_p1, 2.0]);
692      *   var ipex2_p2 = ipex2_board.create('point', [1.0, 1.0]);
693      *   var ipex2_p3 = ipex2_board.create('point', [5.0, 3.0]);
694      *   var ipex2_l1 = ipex2_board.create('line', [ipex2_p2, ipex2_p3]);
695      *   var ipex2_i = ipex2_board.create('intersection', [ipex2_c1, ipex2_l1, 0], {name:'D'});
696      *   var ipex2_j = ipex2_board.create('otherintersection', [ipex2_c1, ipex2_l1, ipex2_i], {name:'E'});
697      * </script><pre>
698      */
699     JXG.createOtherIntersectionPoint = function (board, parents, attributes) {
700         var el, el1, el2, other;
701 
702         if (parents.length !== 3 ||
703                 !Type.isPoint(parents[2]) ||
704                 (parents[0].elementClass !== Const.OBJECT_CLASS_LINE && parents[0].elementClass !== Const.OBJECT_CLASS_CIRCLE) ||
705                 (parents[1].elementClass !== Const.OBJECT_CLASS_LINE && parents[1].elementClass !== Const.OBJECT_CLASS_CIRCLE)) {
706             // Failure
707             throw new Error("JSXGraph: Can't create 'other intersection point' with parent types '" +
708                 (typeof parents[0]) + "',  '" + (typeof parents[1]) + "'and  '" + (typeof parents[2]) + "'." +
709                 "\nPossible parent types: [circle|line,circle|line,point]");
710         }
711 
712         el1 = board.select(parents[0]);
713         el2 = board.select(parents[1]);
714         other = board.select(parents[2]);
715 
716         el = board.create('point', [function () {
717             var c = Geometry.meet(el1.stdform, el2.stdform, 0, el1.board);
718 
719             if (Math.abs(other.X() - c.usrCoords[1]) > Mat.eps ||
720                     Math.abs(other.Y() - c.usrCoords[2]) > Mat.eps ||
721                     Math.abs(other.Z() - c.usrCoords[0]) > Mat.eps) {
722                 return c;
723             }
724 
725             return Geometry.meet(el1.stdform, el2.stdform, 1, el1.board);
726         }], attributes);
727 
728         el.type = Const.OBJECT_TYPE_INTERSECTION;
729         el.elType = 'otherintersection';
730         el.setParents([el1.id, el2.id, other]);
731 
732         el1.addChild(el);
733         el2.addChild(el);
734 
735         el.generatePolynomial = function () {
736             var poly1 = el1.generatePolynomial(el),
737                 poly2 = el2.generatePolynomial(el);
738 
739             if ((poly1.length === 0) || (poly2.length === 0)) {
740                 return [];
741             }
742 
743             return [poly1[0], poly2[0]];
744         };
745 
746         return el;
747     };
748 
749     /**
750      * @class This element is used to provide a constructor for the pole point of a line with respect to a conic or a circle.
751      * @pseudo
752      * @description The pole point is the unique reciprocal relationship of a line with respect to a conic.
753      * The lines tangent to the intersections of a conic and a line intersect at the pole point of that line with respect to that conic.
754      * A line tangent to a conic has the pole point of that line with respect to that conic as the tangent point.
755      * See {@link http://en.wikipedia.org/wiki/Pole_and_polar} for more information on pole and polar.
756      * @name PolePoint
757      * @augments JXG.Point
758      * @constructor
759      * @type JXG.Point
760      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
761      * @param {JXG.Conic,JXG.Circle_JXG.Point} el1,el2 or
762      * @param {JXG.Point_JXG.Conic,JXG.Circle} el1,el2 The result will be the pole point of the line with respect to the conic or the circle.
763      * @example
764      * // Create the pole point of a line with respect to a conic
765      * var p1 = board.create('point', [-1, 2]);
766      * var p2 = board.create('point', [ 1, 4]);
767      * var p3 = board.create('point', [-1,-2]);
768      * var p4 = board.create('point', [ 0, 0]);
769      * var p5 = board.create('point', [ 4,-2]);
770      * var c1 = board.create('conic',[p1,p2,p3,p4,p5]);
771      * var p6 = board.create('point', [-1, 4]);
772      * var p7 = board.create('point', [2, -2]);
773      * var l1 = board.create('line', [p6, p7]);
774      * var p8 = board.create('polepoint', [c1, l1]);
775      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-8018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
776      * <script type='text/javascript'>
777      * var ppex1_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-8018d0d17a98', {boundingbox: [-3, 5, 5, -3], axis: true, showcopyright: false, shownavigation: false});
778      * var ppex1_p1 = ppex1_board.create('point', [-1, 2]);
779      * var ppex1_p2 = ppex1_board.create('point', [ 1, 4]);
780      * var ppex1_p3 = ppex1_board.create('point', [-1,-2]);
781      * var ppex1_p4 = ppex1_board.create('point', [ 0, 0]);
782      * var ppex1_p5 = ppex1_board.create('point', [ 4,-2]);
783      * var ppex1_c1 = ppex1_board.create('conic',[ppex1_p1,ppex1_p2,ppex1_p3,ppex1_p4,ppex1_p5]);
784      * var ppex1_p6 = ppex1_board.create('point', [-1, 4]);
785      * var ppex1_p7 = ppex1_board.create('point', [2, -2]);
786      * var ppex1_l1 = ppex1_board.create('line', [ppex1_p6, ppex1_p7]);
787      * var ppex1_p8 = ppex1_board.create('polepoint', [ppex1_c1, ppex1_l1]);
788      * </script><pre>
789      * @example
790      * // Create the pole point of a line with respect to a circle
791      * var p1 = board.create('point', [1, 1]);
792      * var p2 = board.create('point', [2, 3]);
793      * var c1 = board.create('circle',[p1,p2]);
794      * var p3 = board.create('point', [-1, 4]);
795      * var p4 = board.create('point', [4, -1]);
796      * var l1 = board.create('line', [p3, p4]);
797      * var p5 = board.create('polepoint', [c1, l1]);
798      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-9018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
799      * <script type='text/javascript'>
800      * var ppex2_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-9018d0d17a98', {boundingbox: [-3, 7, 7, -3], axis: true, showcopyright: false, shownavigation: false});
801      * var ppex2_p1 = ppex2_board.create('point', [1, 1]);
802      * var ppex2_p2 = ppex2_board.create('point', [2, 3]);
803      * var ppex2_c1 = ppex2_board.create('circle',[ppex2_p1,ppex2_p2]);
804      * var ppex2_p3 = ppex2_board.create('point', [-1, 4]);
805      * var ppex2_p4 = ppex2_board.create('point', [4, -1]);
806      * var ppex2_l1 = ppex2_board.create('line', [ppex2_p3, ppex2_p4]);
807      * var ppex2_p5 = ppex2_board.create('polepoint', [ppex2_c1, ppex2_l1]);
808      * </script><pre>
809      */
810     JXG.createPolePoint = function (board, parents, attributes) {
811         var el, el1, el2,
812             firstParentIsConic, secondParentIsConic,
813             firstParentIsLine, secondParentIsLine;
814 
815         if (parents.length > 1) {
816             firstParentIsConic = (parents[0].type === Const.OBJECT_TYPE_CONIC ||
817                 parents[0].elementClass === Const.OBJECT_CLASS_CIRCLE);
818             secondParentIsConic = (parents[1].type === Const.OBJECT_TYPE_CONIC ||
819                 parents[1].elementClass === Const.OBJECT_CLASS_CIRCLE);
820 
821             firstParentIsLine = (parents[0].elementClass === Const.OBJECT_CLASS_LINE);
822             secondParentIsLine = (parents[1].elementClass === Const.OBJECT_CLASS_LINE);
823         }
824 
825 /*        if (parents.length !== 2 || !((
826                 parents[0].type === Const.OBJECT_TYPE_CONIC ||
827                 parents[0].elementClass === Const.OBJECT_CLASS_CIRCLE) &&
828                 parents[1].elementClass === Const.OBJECT_CLASS_LINE ||
829                 parents[0].elementClass === Const.OBJECT_CLASS_LINE && (
830                 parents[1].type === Const.OBJECT_TYPE_CONIC ||
831                 parents[1].elementClass === Const.OBJECT_CLASS_CIRCLE))) {*/
832         if (parents.length !== 2 ||
833                 !((firstParentIsConic && secondParentIsLine) ||
834                     (firstParentIsLine && secondParentIsConic))) {
835             // Failure
836             throw new Error("JSXGraph: Can't create 'pole point' with parent types '" +
837                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
838                 "\nPossible parent type: [conic|circle,line], [line,conic|circle]");
839         }
840 
841         if (secondParentIsLine) {
842             el1 = board.select(parents[0]);
843             el2 = board.select(parents[1]);
844         } else {
845             el1 = board.select(parents[1]);
846             el2 = board.select(parents[0]);
847         }
848 
849         el = board.create('point',
850             [function () {
851                 var q = el1.quadraticform,
852                     s = el2.stdform.slice(0, 3);
853 
854                 return [JXG.Math.Numerics.det([s, q[1], q[2]]),
855                         JXG.Math.Numerics.det([q[0], s, q[2]]),
856                         JXG.Math.Numerics.det([q[0], q[1], s])];
857             }], attributes);
858 
859         el.elType = 'polepoint';
860         el.setParents([el1.id, el2.id]);
861 
862         el1.addChild(el);
863         el2.addChild(el);
864 
865         return el;
866     };
867 
868     JXG.registerElement('point', JXG.createPoint);
869     JXG.registerElement('glider', JXG.createGlider);
870     JXG.registerElement('intersection', JXG.createIntersectionPoint);
871     JXG.registerElement('otherintersection', JXG.createOtherIntersectionPoint);
872     JXG.registerElement('polepoint', JXG.createPolePoint);
873 
874     return {
875         Point: JXG.Point,
876         createPoint: JXG.createPoint,
877         createGlider: JXG.createGlider,
878         createIntersection: JXG.createIntersectionPoint,
879         createOtherIntersection: JXG.createOtherIntersectionPoint,
880         createPolePoint: JXG.createPolePoint
881     };
882 });
883