1 /*
  2     Copyright 2008-2021
  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, invMat;
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                 // Case of board.create('point',[baseElement, transform]);
171                 // if (!fromParent) {
172                 //     // The element has been dragged or it is the initial update,
173                 //     // now we transform the baseElement
174                 //     if (this.draggable() && this.baseElement.draggable()) {
175                 //         this.transformations[0].update();
176                 //         invMat = Mat.inverse(this.transformations[0].matrix);
177                 //         c = Mat.matVecMult(invMat, this.coords.usrCoords);
178                 //         this.baseElement.coords.setCoordinates(Const.COORDS_BY_USER, c);
179                 //     }
180                 // }
181                 c = this.transformations[0].apply(this.baseElement);
182             }
183             this.coords.setCoordinates(Const.COORDS_BY_USER, c);
184 
185             for (i = 1; i < this.transformations.length; i++) {
186                 this.coords.setCoordinates(Const.COORDS_BY_USER, this.transformations[i].apply(this));
187             }
188             return this;
189         },
190 
191         /**
192          * Calls the renderer to update the drawing.
193          * @private
194          */
195         updateRenderer: function () {
196             this.updateRendererGeneric('updatePoint');
197             return this;
198         },
199 
200         // documented in JXG.GeometryElement
201         bounds: function () {
202             return this.coords.usrCoords.slice(1).concat(this.coords.usrCoords.slice(1));
203         },
204 
205         /**
206          * Convert the point to intersection point and update the construction.
207          * To move the point visual onto the intersection, a call of board update is necessary.
208          *
209          * @param {String|Object} el1, el2, i, j The intersecting objects and the numbers.
210          **/
211         makeIntersection: function (el1, el2, i, j) {
212             var func;
213 
214             el1 = this.board.select(el1);
215             el2 = this.board.select(el2);
216 
217             func = Geometry.intersectionFunction(this.board, el1, el2, i, j,
218                     Type.evaluate(this.visProp.alwaysintersect));
219             this.addConstraint([func]);
220 
221             try {
222                 el1.addChild(this);
223                 el2.addChild(this);
224             } catch (e) {
225                 throw new Error("JSXGraph: Can't create 'intersection' with parent types '" +
226                     (typeof el1) + "' and '" + (typeof el2) + "'.");
227             }
228 
229             this.type = Const.OBJECT_TYPE_INTERSECTION;
230             this.elType = 'intersection';
231             this.parents = [el1.id, el2.id, i, j];
232 
233             this.generatePolynomial = function () {
234                 var poly1 = el1.generatePolynomial(this),
235                     poly2 = el2.generatePolynomial(this);
236 
237                 if ((poly1.length === 0) || (poly2.length === 0)) {
238                     return [];
239                 }
240 
241                 return [poly1[0], poly2[0]];
242             };
243 
244             this.prepareUpdate().update();
245         },
246 
247         /**
248          * Set the style of a point.
249          * Used for GEONExT import and should not be used to set the point's face and size.
250          * @param {Number} i Integer to determine the style.
251          * @private
252          */
253         setStyle: function (i) {
254             var facemap = [
255                 // 0-2
256                 'cross', 'cross', 'cross',
257                 // 3-6
258                 'circle', 'circle', 'circle', 'circle',
259                 // 7-9
260                 'square', 'square', 'square',
261                 // 10-12
262                 'plus', 'plus', 'plus'
263             ], sizemap = [
264                 // 0-2
265                 2, 3, 4,
266                 // 3-6
267                 1, 2, 3, 4,
268                 // 7-9
269                 2, 3, 4,
270                 // 10-12
271                 2, 3, 4
272             ];
273 
274             this.visProp.face = facemap[i];
275             this.visProp.size = sizemap[i];
276 
277             this.board.renderer.changePointStyle(this);
278             return this;
279         },
280 
281         /**
282          * @deprecated Use JXG#normalizePointFace instead
283          * @param s
284          * @returns {*}
285          */
286         normalizeFace: function (s) {
287             JXG.deprecated('Point.normalizeFace()', 'JXG.normalizePointFace()');
288             return Options.normalizePointFace(s);
289         },
290 
291         /**
292          * Set the face of a point element.
293          * @param {String} f String which determines the face of the point. See {@link JXG.GeometryElement#face} for a list of available faces.
294          * @see JXG.GeometryElement#face
295          * @deprecated Use setAttribute()
296          */
297         face: function (f) {
298             JXG.deprecated('Point.face()', 'Point.setAttribute()');
299             this.setAttribute({face: f});
300         },
301 
302         /**
303          * Set the size of a point element
304          * @param {Number} s Integer which determines the size of the point.
305          * @see JXG.GeometryElement#size
306          * @deprecated Use setAttribute()
307          */
308         size: function (s) {
309             JXG.deprecated('Point.size()', 'Point.setAttribute()');
310             this.setAttribute({size: s});
311         },
312 
313         /**
314          * Test if the point is on (is incident with) element "el".
315          *
316          * @param {JXG.GeometryElement} el
317          * @param {Number} tol
318          * @returns {Boolean}
319          *
320          * @example
321          * var circ = board.create('circle', [[-2, -2], 1]);
322          * var seg = board.create('segment', [[-1, -3], [0,0]]);
323          * var line = board.create('line', [[1, 3], [2, -2]]);
324          * var po = board.create('point', [-1, 0], {color: 'blue'});
325          * var curve = board.create('functiongraph', ['sin(x)'], {strokeColor: 'blue'});
326          * var pol = board.create('polygon', [[2,2], [4,2], [4,3]], {strokeColor: 'blue'});
327          *
328          * var point = board.create('point', [-1, 1], {
329          *               attractors: [line, seg, circ, po, curve, pol],
330          *               attractorDistance: 0.2
331          *             });
332          *
333          * var txt = board.create('text', [-4, 3, function() {
334          *              return 'point on line: ' + point.isOn(line) + '<br>' +
335          *                 'point on seg: ' + point.isOn(seg) + '<br>' +
336          *                 'point on circ = ' + point.isOn(circ) + '<br>' +
337          *                 'point on point = ' + point.isOn(po) + '<br>' +
338          *                 'point on curve = ' + point.isOn(curve) + '<br>' +
339          *                 'point on polygon = ' + point.isOn(pol) + '<br>';
340          * }]);
341          *
342          * </pre><div id="JXG6c7d7404-758a-44eb-802c-e9644b9fab71" class="jxgbox" style="width: 300px; height: 300px;"></div>
343          * <script type="text/javascript">
344          *     (function() {
345          *         var board = JXG.JSXGraph.initBoard('JXG6c7d7404-758a-44eb-802c-e9644b9fab71',
346          *             {boundingbox: [-8, 8, 8,-8], axis: true, showcopyright: false, shownavigation: false});
347          *     var circ = board.create('circle', [[-2, -2], 1]);
348          *     var seg = board.create('segment', [[-1, -3], [0,0]]);
349          *     var line = board.create('line', [[1, 3], [2, -2]]);
350          *     var po = board.create('point', [-1, 0], {color: 'blue'});
351          *     var curve = board.create('functiongraph', ['sin(x)'], {strokeColor: 'blue'});
352          *     var pol = board.create('polygon', [[2,2], [4,2], [4,3]], {strokeColor: 'blue'});
353          *
354          *     var point = board.create('point', [-1, 1], {
355          *     			  attractors: [line, seg, circ, po, curve, pol],
356          *                   attractorDistance: 0.2
357          *                 });
358          *
359          *     var txt = board.create('text', [-4, 3, function() {
360          *     		return 'point on line: ' + point.isOn(line) + '<br>' +
361          *                     'point on seg: ' + point.isOn(seg) + '<br>' +
362          *                     'point on circ = ' + point.isOn(circ) + '<br>' +
363          *                     'point on point = ' + point.isOn(po) + '<br>' +
364          *                     'point on curve = ' + point.isOn(curve) + '<br>' +
365          *                     'point on polygon = ' + point.isOn(pol) + '<br>';
366          *     }]);
367          *
368          *     })();
369          *
370          * </script><pre>
371          *
372          */
373         isOn: function(el, tol) {
374             var arr, crds;
375 
376             tol = tol || Mat.eps;
377 
378             if (Type.isPoint(el)) {
379                 return this.Dist(el) < tol;
380             } else if (el.elementClass === Const.OBJECT_CLASS_LINE) {
381                 if (el.elType === 'segment' && !Type.evaluate(this.visProp.alwaysintersect)) {
382                     arr = JXG.Math.Geometry.projectCoordsToSegment(
383             			        this.coords.usrCoords,
384                                 el.point1.coords.usrCoords,
385                                 el.point2.coords.usrCoords);
386                     if (arr[1] >= 0 && arr[1] <= 1 &&
387                         Geometry.distPointLine(this.coords.usrCoords, el.stdform) < tol) {
388        				    return true;
389                     } else {
390             		    return false;
391                     }
392                 } else {
393                     return Geometry.distPointLine(this.coords.usrCoords, el.stdform) < tol;
394                 }
395             } else if (el.elementClass === Const.OBJECT_CLASS_CIRCLE) {
396                 if (Type.evaluate(el.visProp.hasinnerpoints)) {
397                     return this.Dist(el.center) < el.Radius() + tol;
398                 }
399                 return Math.abs(this.Dist(el.center) - el.Radius()) < tol;
400             } else if (el.elementClass === Const.OBJECT_CLASS_CURVE) {
401                 crds = Geometry.projectPointToCurve(this, el, this.board)[0];
402                 return Geometry.distance(this.coords.usrCoords, crds.usrCoords, 3) < tol;
403             } else if (el.type === Const.OBJECT_TYPE_POLYGON) {
404                 if (Type.evaluate(el.visProp.hasinnerpoints)) {
405                     if (el.pnpoly(this.coords.usrCoords[1], this.coords.usrCoords[2], JXG.COORDS_BY_USER)) {
406                         return true;
407                     }
408                 }
409                 arr = Geometry.projectCoordsToPolygon(this.coords.usrCoords, el);
410                 return Geometry.distance(this.coords.usrCoords, arr, 3) < tol;
411             } else if (el.type === Const.OBJECT_TYPE_TURTLE) {
412                 crds = Geometry.projectPointToTurtle(this, el, this.board);
413                 return Geometry.distance(this.coords.usrCoords, crds.usrCoords, 3) < tol;
414             }
415 
416             // TODO: Arc, Sector
417             return false;
418         },
419 
420         // Already documented in GeometryElement
421         cloneToBackground: function () {
422             var copy = {};
423 
424             copy.id = this.id + 'T' + this.numTraces;
425             this.numTraces += 1;
426 
427             copy.coords = this.coords;
428             copy.visProp = Type.deepCopy(this.visProp, this.visProp.traceattributes, true);
429             copy.visProp.layer = this.board.options.layer.trace;
430             copy.elementClass = Const.OBJECT_CLASS_POINT;
431             copy.board = this.board;
432             Type.clearVisPropOld(copy);
433 
434             copy.visPropCalc = {
435                 visible: Type.evaluate(copy.visProp.visible)
436             };
437 
438             this.board.renderer.drawPoint(copy);
439             this.traces[copy.id] = copy.rendNode;
440 
441             return this;
442         }
443 
444     });
445 
446     /**
447      * @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
448      * 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
449      * 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
450      * position directly.
451      * @pseudo
452      * @description
453      * @name Point
454      * @augments JXG.Point
455      * @constructor
456      * @type JXG.Point
457      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
458      * @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
459      * constraint, or a function which takes no parameter and returns a number. Every parent element determines one coordinate. If a coordinate is
460      * 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
461      * 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
462      * 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
463      * parent elements are given they will be interpreted as homogeneous coordinates.
464      * @param {JXG.Point_JXG.Transformation_Array} Point,Transformation A point can also be created providing a transformation or an array of transformations.
465      * The resulting point is a clone of the base point transformed by the given Transformation. {@see JXG.Transformation}.
466      *
467      * @example
468      * // Create a free point using affine Euclidean coordinates
469      * var p1 = board.create('point', [3.5, 2.0]);
470      * </pre><div class="jxgbox" id="JXG672f1764-7dfa-4abc-a2c6-81fbbf83e44b" style="width: 200px; height: 200px;"></div>
471      * <script type="text/javascript">
472      *   var board = JXG.JSXGraph.initBoard('JXG672f1764-7dfa-4abc-a2c6-81fbbf83e44b', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
473      *   var p1 = board.create('point', [3.5, 2.0]);
474      * </script><pre>
475      * @example
476      * // Create a constrained point using anonymous function
477      * var p2 = board.create('point', [3.5, function () { return p1.X(); }]);
478      * </pre><div class="jxgbox" id="JXG4fd4410c-3383-4e80-b1bb-961f5eeef224" style="width: 200px; height: 200px;"></div>
479      * <script type="text/javascript">
480      *   var fpex1_board = JXG.JSXGraph.initBoard('JXG4fd4410c-3383-4e80-b1bb-961f5eeef224', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
481      *   var fpex1_p1 = fpex1_board.create('point', [3.5, 2.0]);
482      *   var fpex1_p2 = fpex1_board.create('point', [3.5, function () { return fpex1_p1.X(); }]);
483      * </script><pre>
484      * @example
485      * // Create a point using transformations
486      * var trans = board.create('transform', [2, 0.5], {type:'scale'});
487      * var p3 = board.create('point', [p2, trans]);
488      * </pre><div class="jxgbox" id="JXG630afdf3-0a64-46e0-8a44-f51bd197bb8d" style="width: 400px; height: 400px;"></div>
489      * <script type="text/javascript">
490      *   var fpex2_board = JXG.JSXGraph.initBoard('JXG630afdf3-0a64-46e0-8a44-f51bd197bb8d', {boundingbox: [-1, 9, 9, -1], axis: true, showcopyright: false, shownavigation: false});
491      *   var fpex2_trans = fpex2_board.create('transform', [2, 0.5], {type:'scale'});
492      *   var fpex2_p2 = fpex2_board.create('point', [3.5, 2.0]);
493      *   var fpex2_p3 = fpex2_board.create('point', [fpex2_p2, fpex2_trans]);
494      * </script><pre>
495      */
496     JXG.createPoint = function (board, parents, attributes) {
497         var el, attr;
498 
499         attr = Type.copyAttributes(attributes, board.options, 'point');
500         el = CoordsElement.create(JXG.Point, board, parents, attr);
501         if (!el) {
502             throw new Error("JSXGraph: Can't create point with parent types '" +
503                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
504                     "\nPossible parent types: [x,y], [z,x,y], [element,transformation]");
505         }
506 
507         return el;
508     };
509 
510     /**
511      * @class This element is used to provide a constructor for a glider point.
512      * @pseudo
513      * @description A glider is a point which lives on another geometric element like a line, circle, curve, turtle.
514      * @name Glider
515      * @augments JXG.Point
516      * @constructor
517      * @type JXG.Point
518      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
519      * @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.
520      * 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
521      * coordinates, otherwise they will be interpreted as homogeneous coordinates. In any case the point will be projected on the glide object.
522      * @example
523      * // Create a glider with user defined coordinates. If the coordinates are not on
524      * // the circle (like in this case) the point will be projected onto the circle.
525      * var p1 = board.create('point', [2.0, 2.0]);
526      * var c1 = board.create('circle', [p1, 2.0]);
527      * var p2 = board.create('glider', [2.0, 1.5, c1]);
528      * </pre><div class="jxgbox" id="JXG4f65f32f-e50a-4b50-9b7c-f6ec41652930" style="width: 300px; height: 300px;"></div>
529      * <script type="text/javascript">
530      *   var gpex1_board = JXG.JSXGraph.initBoard('JXG4f65f32f-e50a-4b50-9b7c-f6ec41652930', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
531      *   var gpex1_p1 = gpex1_board.create('point', [2.0, 2.0]);
532      *   var gpex1_c1 = gpex1_board.create('circle', [gpex1_p1, 2.0]);
533      *   var gpex1_p2 = gpex1_board.create('glider', [2.0, 1.5, gpex1_c1]);
534      * </script><pre>
535      * @example
536      * // Create a glider with default coordinates (1,0,0). Same premises as above.
537      * var p1 = board.create('point', [2.0, 2.0]);
538      * var c1 = board.create('circle', [p1, 2.0]);
539      * var p2 = board.create('glider', [c1]);
540      * </pre><div class="jxgbox" id="JXG4de7f181-631a-44b1-a12f-bc4d995609e8" style="width: 200px; height: 200px;"></div>
541      * <script type="text/javascript">
542      *   var gpex2_board = JXG.JSXGraph.initBoard('JXG4de7f181-631a-44b1-a12f-bc4d995609e8', {boundingbox: [-1, 5, 5, -1], axis: true, showcopyright: false, shownavigation: false});
543      *   var gpex2_p1 = gpex2_board.create('point', [2.0, 2.0]);
544      *   var gpex2_c1 = gpex2_board.create('circle', [gpex2_p1, 2.0]);
545      *   var gpex2_p2 = gpex2_board.create('glider', [gpex2_c1]);
546      * </script><pre>
547      *@example
548      * //animate example 2
549      * var p1 = board.create('point', [2.0, 2.0]);
550      * var c1 = board.create('circle', [p1, 2.0]);
551      * var p2 = board.create('glider', [c1]);
552      * var button1 = board.create('button', [1, 7, 'start animation',function(){p2.startAnimation(1,4)}]);
553      * var button2 = board.create('button', [1, 5, 'stop animation',function(){p2.stopAnimation()}]);
554      * </pre><div class="jxgbox" id="JXG4de7f181-631a-44b1-a12f-bc4d133709e8" style="width: 200px; height: 200px;"></div>
555      * <script type="text/javascript">
556      *   var gpex3_board = JXG.JSXGraph.initBoard('JXG4de7f181-631a-44b1-a12f-bc4d133709e8', {boundingbox: [-1, 10, 10, -1], axis: true, showcopyright: false, shownavigation: false});
557      *   var gpex3_p1 = gpex3_board.create('point', [2.0, 2.0]);
558      *   var gpex3_c1 = gpex3_board.create('circle', [gpex3_p1, 2.0]);
559      *   var gpex3_p2 = gpex3_board.create('glider', [gpex3_c1]);
560      *   gpex3_board.create('button', [1, 7, 'start animation',function(){gpex3_p2.startAnimation(1,4)}]);
561      *   gpex3_board.create('button', [1, 5, 'stop animation',function(){gpex3_p2.stopAnimation()}]);
562      * </script><pre>
563      */
564     JXG.createGlider = function (board, parents, attributes) {
565         var el, coords,
566             attr = Type.copyAttributes(attributes, board.options, 'glider');
567 
568         if (parents.length === 1) {
569             coords = [0, 0];
570         } else {
571             coords = parents.slice(0, 2);
572         }
573         el = board.create('point', coords, attr);
574 
575         // eltype is set in here
576         el.makeGlider(parents[parents.length - 1]);
577 
578         return el;
579     };
580 
581 
582     /**
583      * @class An intersection point is a point which lives on two JSXGraph elements, i.e. it is one point of the the set
584      * consisting of the intersection points of the two elements. The following element types can be (mutually) intersected: line, circle,
585      * curve, polygon, polygonal chain.
586      *
587      * @pseudo
588      * @name Intersection
589      * @augments JXG.Point
590      * @constructor
591      * @type JXG.Point
592      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
593      * @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
594      * intersection point if two points are available: <ul>
595      *   <li>i==0: use the positive square root,</li>
596      *   <li>i==1: use the negative square root.</li></ul>
597      * @example
598      * // Create an intersection point of circle and line
599      * var p1 = board.create('point', [2.0, 2.0]);
600      * var c1 = board.create('circle', [p1, 2.0]);
601      *
602      * var p2 = board.create('point', [2.0, 2.0]);
603      * var p3 = board.create('point', [2.0, 2.0]);
604      * var l1 = board.create('line', [p2, p3]);
605      *
606      * var i = board.create('intersection', [c1, l1, 0]);
607      * </pre><div class="jxgbox" id="JXGe5b0e190-5200-4bc3-b995-b6cc53dc5dc0" style="width: 300px; height: 300px;"></div>
608      * <script type="text/javascript">
609      *   var ipex1_board = JXG.JSXGraph.initBoard('JXGe5b0e190-5200-4bc3-b995-b6cc53dc5dc0', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
610      *   var ipex1_p1 = ipex1_board.create('point', [4.0, 4.0]);
611      *   var ipex1_c1 = ipex1_board.create('circle', [ipex1_p1, 2.0]);
612      *   var ipex1_p2 = ipex1_board.create('point', [1.0, 1.0]);
613      *   var ipex1_p3 = ipex1_board.create('point', [5.0, 3.0]);
614      *   var ipex1_l1 = ipex1_board.create('line', [ipex1_p2, ipex1_p3]);
615      *   var ipex1_i = ipex1_board.create('intersection', [ipex1_c1, ipex1_l1, 0]);
616      * </script><pre>
617      */
618     JXG.createIntersectionPoint = function (board, parents, attributes) {
619         var el, el1, el2, func, i, j,
620             attr = Type.copyAttributes(attributes, board.options, 'intersection');
621 
622         // make sure we definitely have the indices
623         parents.push(0, 0);
624 
625         el1 = board.select(parents[0]);
626         el2 = board.select(parents[1]);
627 
628         i = parents[2] || 0;
629         j = parents[3] || 0;
630 
631         el = board.create('point', [0, 0, 0], attr);
632 
633         // el.visProp.alwaysintersect is evaluated as late as in the returned function
634         func = Geometry.intersectionFunction(board, el1, el2, i, j, el.visProp.alwaysintersect);
635         el.addConstraint([func]);
636 
637         try {
638             el1.addChild(el);
639             el2.addChild(el);
640         } catch (e) {
641             throw new Error("JSXGraph: Can't create 'intersection' with parent types '" +
642                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'.");
643         }
644 
645         el.type = Const.OBJECT_TYPE_INTERSECTION;
646         el.elType = 'intersection';
647         el.setParents([el1.id, el2.id]);
648 
649         /**
650          * Array of length 2 containing the numbers i and j.
651          * The intersection point is i-th intersection point.
652          * j is unused.
653          * @type Array
654          * @private
655          */
656         el.intersectionNumbers = [i, j];
657         el.getParents = function() {
658             return this.parents.concat(this.intersectionNumbers);
659         };
660 
661         el.generatePolynomial = function () {
662             var poly1 = el1.generatePolynomial(el),
663                 poly2 = el2.generatePolynomial(el);
664 
665             if ((poly1.length === 0) || (poly2.length === 0)) {
666                 return [];
667             }
668 
669             return [poly1[0], poly2[0]];
670         };
671 
672         return el;
673     };
674 
675     /**
676      * @class This element is used to provide a constructor for the "other" intersection point.
677      * @pseudo
678      * @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.
679      * an intersection point of the two elements. Additionally, one intersection point is provided. The function returns the other intersection point.
680      * @name OtherIntersection
681      * @augments JXG.Point
682      * @constructor
683      * @type JXG.Point
684      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
685      * @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
686      * intersection point different from p:
687      * @example
688      * // Create an intersection point of circle and line
689      * var p1 = board.create('point', [2.0, 2.0]);
690      * var c1 = board.create('circle', [p1, 2.0]);
691      *
692      * var p2 = board.create('point', [2.0, 2.0]);
693      * var p3 = board.create('point', [2.0, 2.0]);
694      * var l1 = board.create('line', [p2, p3]);
695      *
696      * var i = board.create('intersection', [c1, l1, 0]);
697      * var j = board.create('otherintersection', [c1, l1, i]);
698      * </pre><div class="jxgbox" id="JXG45e25f12-a1de-4257-a466-27a2ae73614c" style="width: 300px; height: 300px;"></div>
699      * <script type="text/javascript">
700      *   var ipex2_board = JXG.JSXGraph.initBoard('JXG45e25f12-a1de-4257-a466-27a2ae73614c', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
701      *   var ipex2_p1 = ipex2_board.create('point', [4.0, 4.0]);
702      *   var ipex2_c1 = ipex2_board.create('circle', [ipex2_p1, 2.0]);
703      *   var ipex2_p2 = ipex2_board.create('point', [1.0, 1.0]);
704      *   var ipex2_p3 = ipex2_board.create('point', [5.0, 3.0]);
705      *   var ipex2_l1 = ipex2_board.create('line', [ipex2_p2, ipex2_p3]);
706      *   var ipex2_i = ipex2_board.create('intersection', [ipex2_c1, ipex2_l1, 0], {name:'D'});
707      *   var ipex2_j = ipex2_board.create('otherintersection', [ipex2_c1, ipex2_l1, ipex2_i], {name:'E'});
708      * </script><pre>
709      */
710     JXG.createOtherIntersectionPoint = function (board, parents, attributes) {
711         var el, el1, el2, other;
712 
713         if (parents.length !== 3 ||
714                 !Type.isPoint(parents[2]) ||
715                 (parents[0].elementClass !== Const.OBJECT_CLASS_LINE && parents[0].elementClass !== Const.OBJECT_CLASS_CIRCLE) ||
716                 (parents[1].elementClass !== Const.OBJECT_CLASS_LINE && parents[1].elementClass !== Const.OBJECT_CLASS_CIRCLE)) {
717             // Failure
718             throw new Error("JSXGraph: Can't create 'other intersection point' with parent types '" +
719                 (typeof parents[0]) + "',  '" + (typeof parents[1]) + "'and  '" + (typeof parents[2]) + "'." +
720                 "\nPossible parent types: [circle|line,circle|line,point]");
721         }
722 
723         el1 = board.select(parents[0]);
724         el2 = board.select(parents[1]);
725         other = board.select(parents[2]);
726 
727         el = board.create('point', [function () {
728             var c = Geometry.meet(el1.stdform, el2.stdform, 0, el1.board);
729 
730             if (Math.abs(other.X() - c.usrCoords[1]) > Mat.eps ||
731                     Math.abs(other.Y() - c.usrCoords[2]) > Mat.eps ||
732                     Math.abs(other.Z() - c.usrCoords[0]) > Mat.eps) {
733                 return c;
734             }
735 
736             return Geometry.meet(el1.stdform, el2.stdform, 1, el1.board);
737         }], attributes);
738 
739         el.type = Const.OBJECT_TYPE_INTERSECTION;
740         el.elType = 'otherintersection';
741         el.setParents([el1.id, el2.id, other]);
742 
743         el1.addChild(el);
744         el2.addChild(el);
745 
746         el.generatePolynomial = function () {
747             var poly1 = el1.generatePolynomial(el),
748                 poly2 = el2.generatePolynomial(el);
749 
750             if ((poly1.length === 0) || (poly2.length === 0)) {
751                 return [];
752             }
753 
754             return [poly1[0], poly2[0]];
755         };
756 
757         return el;
758     };
759 
760     /**
761      * @class This element is used to provide a constructor for the pole point of a line with respect to a conic or a circle.
762      * @pseudo
763      * @description The pole point is the unique reciprocal relationship of a line with respect to a conic.
764      * 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.
765      * A line tangent to a conic has the pole point of that line with respect to that conic as the tangent point.
766      * See {@link http://en.wikipedia.org/wiki/Pole_and_polar} for more information on pole and polar.
767      * @name PolePoint
768      * @augments JXG.Point
769      * @constructor
770      * @type JXG.Point
771      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
772      * @param {JXG.Conic,JXG.Circle_JXG.Point} el1,el2 or
773      * @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.
774      * @example
775      * // Create the pole point of a line with respect to a conic
776      * var p1 = board.create('point', [-1, 2]);
777      * var p2 = board.create('point', [ 1, 4]);
778      * var p3 = board.create('point', [-1,-2]);
779      * var p4 = board.create('point', [ 0, 0]);
780      * var p5 = board.create('point', [ 4,-2]);
781      * var c1 = board.create('conic',[p1,p2,p3,p4,p5]);
782      * var p6 = board.create('point', [-1, 4]);
783      * var p7 = board.create('point', [2, -2]);
784      * var l1 = board.create('line', [p6, p7]);
785      * var p8 = board.create('polepoint', [c1, l1]);
786      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-8018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
787      * <script type='text/javascript'>
788      * var ppex1_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-8018d0d17a98', {boundingbox: [-3, 5, 5, -3], axis: true, showcopyright: false, shownavigation: false});
789      * var ppex1_p1 = ppex1_board.create('point', [-1, 2]);
790      * var ppex1_p2 = ppex1_board.create('point', [ 1, 4]);
791      * var ppex1_p3 = ppex1_board.create('point', [-1,-2]);
792      * var ppex1_p4 = ppex1_board.create('point', [ 0, 0]);
793      * var ppex1_p5 = ppex1_board.create('point', [ 4,-2]);
794      * var ppex1_c1 = ppex1_board.create('conic',[ppex1_p1,ppex1_p2,ppex1_p3,ppex1_p4,ppex1_p5]);
795      * var ppex1_p6 = ppex1_board.create('point', [-1, 4]);
796      * var ppex1_p7 = ppex1_board.create('point', [2, -2]);
797      * var ppex1_l1 = ppex1_board.create('line', [ppex1_p6, ppex1_p7]);
798      * var ppex1_p8 = ppex1_board.create('polepoint', [ppex1_c1, ppex1_l1]);
799      * </script><pre>
800      * @example
801      * // Create the pole point of a line with respect to a circle
802      * var p1 = board.create('point', [1, 1]);
803      * var p2 = board.create('point', [2, 3]);
804      * var c1 = board.create('circle',[p1,p2]);
805      * var p3 = board.create('point', [-1, 4]);
806      * var p4 = board.create('point', [4, -1]);
807      * var l1 = board.create('line', [p3, p4]);
808      * var p5 = board.create('polepoint', [c1, l1]);
809      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-9018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
810      * <script type='text/javascript'>
811      * var ppex2_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-9018d0d17a98', {boundingbox: [-3, 7, 7, -3], axis: true, showcopyright: false, shownavigation: false});
812      * var ppex2_p1 = ppex2_board.create('point', [1, 1]);
813      * var ppex2_p2 = ppex2_board.create('point', [2, 3]);
814      * var ppex2_c1 = ppex2_board.create('circle',[ppex2_p1,ppex2_p2]);
815      * var ppex2_p3 = ppex2_board.create('point', [-1, 4]);
816      * var ppex2_p4 = ppex2_board.create('point', [4, -1]);
817      * var ppex2_l1 = ppex2_board.create('line', [ppex2_p3, ppex2_p4]);
818      * var ppex2_p5 = ppex2_board.create('polepoint', [ppex2_c1, ppex2_l1]);
819      * </script><pre>
820      */
821     JXG.createPolePoint = function (board, parents, attributes) {
822         var el, el1, el2,
823             firstParentIsConic, secondParentIsConic,
824             firstParentIsLine, secondParentIsLine;
825 
826         if (parents.length > 1) {
827             firstParentIsConic = (parents[0].type === Const.OBJECT_TYPE_CONIC ||
828                 parents[0].elementClass === Const.OBJECT_CLASS_CIRCLE);
829             secondParentIsConic = (parents[1].type === Const.OBJECT_TYPE_CONIC ||
830                 parents[1].elementClass === Const.OBJECT_CLASS_CIRCLE);
831 
832             firstParentIsLine = (parents[0].elementClass === Const.OBJECT_CLASS_LINE);
833             secondParentIsLine = (parents[1].elementClass === Const.OBJECT_CLASS_LINE);
834         }
835 
836 /*        if (parents.length !== 2 || !((
837                 parents[0].type === Const.OBJECT_TYPE_CONIC ||
838                 parents[0].elementClass === Const.OBJECT_CLASS_CIRCLE) &&
839                 parents[1].elementClass === Const.OBJECT_CLASS_LINE ||
840                 parents[0].elementClass === Const.OBJECT_CLASS_LINE && (
841                 parents[1].type === Const.OBJECT_TYPE_CONIC ||
842                 parents[1].elementClass === Const.OBJECT_CLASS_CIRCLE))) {*/
843         if (parents.length !== 2 ||
844                 !((firstParentIsConic && secondParentIsLine) ||
845                     (firstParentIsLine && secondParentIsConic))) {
846             // Failure
847             throw new Error("JSXGraph: Can't create 'pole point' with parent types '" +
848                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
849                 "\nPossible parent type: [conic|circle,line], [line,conic|circle]");
850         }
851 
852         if (secondParentIsLine) {
853             el1 = board.select(parents[0]);
854             el2 = board.select(parents[1]);
855         } else {
856             el1 = board.select(parents[1]);
857             el2 = board.select(parents[0]);
858         }
859 
860         el = board.create('point',
861             [function () {
862                 var q = el1.quadraticform,
863                     s = el2.stdform.slice(0, 3);
864 
865                 return [JXG.Math.Numerics.det([s, q[1], q[2]]),
866                         JXG.Math.Numerics.det([q[0], s, q[2]]),
867                         JXG.Math.Numerics.det([q[0], q[1], s])];
868             }], attributes);
869 
870         el.elType = 'polepoint';
871         el.setParents([el1.id, el2.id]);
872 
873         el1.addChild(el);
874         el2.addChild(el);
875 
876         return el;
877     };
878 
879     JXG.registerElement('point', JXG.createPoint);
880     JXG.registerElement('glider', JXG.createGlider);
881     JXG.registerElement('intersection', JXG.createIntersectionPoint);
882     JXG.registerElement('otherintersection', JXG.createOtherIntersectionPoint);
883     JXG.registerElement('polepoint', JXG.createPolePoint);
884 
885     return {
886         Point: JXG.Point,
887         createPoint: JXG.createPoint,
888         createGlider: JXG.createGlider,
889         createIntersection: JXG.createIntersectionPoint,
890         createOtherIntersection: JXG.createOtherIntersectionPoint,
891         createPolePoint: JXG.createPolePoint
892     };
893 });
894