-
Notifications
You must be signed in to change notification settings - Fork 2
/
grapher.js
553 lines (446 loc) · 20.5 KB
/
grapher.js
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
// Wait for the DOM to be ready using jQuery.
$(function () {
// Get the 'canvas' DOM element based on its id using jQuery.
var canvas = $('#myCanvas')[0],
// Get the canvas context
c = canvas.getContext('2d'),
/* styling to be fixed later
c.fillStyle = "#e5ff2",
c.opacity = 0.9,
c.fill(),
*/
// n is number of line segments to be graphed
// change based on function being graphed
n,
// Define the user's viewing window, to use when graphing
//Add functionality to change this later
xMin = -20,
xMax = 20,
yMin = -20,
yMax = 20,
// Initialize the Math.js library
math = mathjs(),
// 'expr' will contain the math expression as a string.
expr = '',
// 'scope' defines the variables available inside the math expression.
scope = {
x: 0
},
//expression as a tree
tree;
// Sets the value of 'expr' and re-parses the expression into 'tree'.
function setExpr(newExpr) {
expr = newExpr;
tree = math.parse(expr, scope);
}
//generates axes
function generatePlane() { //generates x and y axes
c.beginPath();
c.strokeStyle = "rgba(124,124,124,0.3)";
var currentPt;
//horizontal(y) lines
for (var i = 0; i <= (yMax - yMin) ; i++) {
currentPt = ((yMin + yMax + i) / (yMax - yMin)) * (canvas.height);
c.moveTo(currentPt, 0);
c.lineTo(currentPt, canvas.height);
}
//vertical(x) lines
for (var i = 0; i <= (xMax - xMin) ; i++) {
currentPt = (xMin + xMax + i) * canvas.width / (xMax - xMin);
c.moveTo(0, currentPt);
c.lineTo(canvas.width, currentPt);
}
c.stroke();
c.beginPath();
//y axis
c.strokeStyle = "#FFF";
c.moveTo(canvas.width / 2, 0);
c.lineTo(canvas.width / 2, canvas.height);
//x axis
c.moveTo(0, canvas.height / 2);
c.lineTo(canvas.width, canvas.height / 2);
c.stroke();
}
generatePlane();
// Plots the math expression curve on the canvas.
function drawCurve(color, current) {
// These variables are used inside the for loop.
var i,
// These vary between xMin and xMax
// and yMin and yMax.
xPixel, yPixel,
// These vary between 0 and 1.
percentX, percentY,
// These are values in math coordinates.
mathX, mathY;
c.save();
// Plot the math expression as a curve using the Canvas API:
// This line of code begins the math curve path definition.
c.beginPath();
// 'n' is the number of points used to define the curve, which
// consists of (n - 1) straight line segments.
for (i = 0; i < n; i++) {
// 'i' varies between 0 and n - 1.
// 'percentX' varies between 0 and 1.
percentX = i / (n - 1);
// 'mathX' varies between 'xMin' and 'xMax'.
mathX = percentX * (xMax - xMin) + xMin;
// mathY = f(mathX)
//if function to graph is f(x)
if (current == 1) {
mathY = evalExpr(mathX);
}
//if function to graph is f'(x)
else if (current == 2) {
mathY = calculateDerivative(mathX);
}
//if function to graph is f"(x)
else if (current == 3) {
mathY = calculateSecondDerivative(mathX);
}
// Maps to canvas coordinates
percentY = (mathY - yMin) / (yMax - yMin);
// Flip Y to match canvas coordinates.
percentY = 1 - percentY;
// Project percentX and percentY to pixel coordinates.
xPixel = percentX * canvas.width;
yPixel = percentY * canvas.height;
c.strokeStyle = color;
// The first time this line of code is run, it defines the first point
// in the path, acting exactly like 'c.moveTo(xPixel, yPixel);'
// Subsequently, this line of code adds a line segment to the curve path
// between the previous and current points.
c.lineTo(xPixel, yPixel);
}
// This line of code renders the curve path defined by the 'c.lineTo'
c.stroke();
c.restore();
}
// Evaluates the current math expression
// Returns a Y coordinate
function evalExpr(mathX) {
// Set values on the scope visible inside the math expression.
scope.x = mathX;
// Evaluate the previously parsed math expression and return it
return tree.eval();
}
//takes approximate derivative at x with alternate form of difference quotient f(x+h)-f(x-h)/2h ~= f'(x)
function calculateDerivative(x) {
//operates on assumption that f(x) = expr
var h = 0.0001;
var derivative = (evalExpr(x + h) - evalExpr(x - h)) / (2 * h); //applied difference quotient
var result = derivative; //Math.round(derivative * 1000000) / 1000000; //rounds answer
//console.log(x, result);
return result;
}
function calculateSecondDerivative(x) {
//takes derivative of the first derivative
var h = 0.0001;
var secondDerivative = (calculateDerivative(x + h) - calculateDerivative(x - h)) / (2 * h); //applied difference quotient
var result = secondDerivative; //Math.round(secondDerivative * 10000000000) / 10000000000; //rounds answer
return result;
}
function extrema() {
var x, y, xVal, yVal, radius;
radius = 10;
var previous = calculateDerivative(xMin - 0.01);
for (var i = xMin; i < xMax; i = i + 0.05) { //change increment value
var derivative = calculateDerivative(i);
if ((derivative < 0 && previous > 0) || (derivative > 0 && previous < 0) || derivative == 0) {
var secondDeriv = calculateSecondDerivative(i);
if (secondDeriv > 0) { //if is rel min
x = Math.round(i * 10) / 10;
y = evalExpr(x);
xVal = (-xMin + x) * canvas.width / (xMax - xMin); //mapped x
yVal = (-yMin + y) * canvas.height / (yMax - yMin); //mapped y
yVal = 600 - yVal; //flips it to match canvas coordinates
//draw circle around this point
c.beginPath();
c.arc(xVal, yVal, radius, 0, 2 * Math.PI, false); //draws circle of radius centered at (xVal, yVal)
c.fillStyle = "rgba(" + $('#hdn5').val() + ",0.5)";
c.fill();
}
else if (secondDeriv < 0) { //if is rel max
x = Math.round(i * 10) / 10;
y = evalExpr(x);
xVal = (-xMin + x) * canvas.width / (xMax - xMin); //mapped x
yVal = (-yMin + y) * canvas.height / (yMax - yMin); //mapped y
yVal = 600 - yVal; //flips it to match canvas coordinates
//draw circle around this point
c.beginPath();
//draws square of radius centered at (xVal, yVal)
c.lineWidth = "2";
c.rect((xVal-10), (yVal-10), 20, 20);
c.fillStyle = "rgba(" + $('#hdn5').val() + ",0.5)";
c.fill();
}
}
var previous = derivative;
}
}
//evaluates second derivative at each x
//if second deriv > 0, colors one color
//if second deriv < 0, colors a different color
//if second deriv has just changed from negative to positive, marks it as inflection point
//takes val of x coordinate and corresponding second derivative as parameter
function concavity() { //assuming f(x) == expr
var xVal;
var width = (0.05) * canvas.width / (xMax - xMin);
var previous = calculateSecondDerivative(xMin - 0.01);
for (var i = xMin; i < xMax; i += 0.01) { //every 0.05
xVal = (-xMin + i) * canvas.width / (xMax - xMin); //mapped x (shifts over + multiplies w/ proportions)
var secondDeriv = calculateSecondDerivative(i); //secondDerivative at x=i
//console.log(i, secondDeriv);
if ((secondDeriv < 0 && previous < 0) || (secondDeriv > 0 && previous > 0)) {
if (secondDeriv < 0) {
//if concave down
//draws rectangle of width 1 and canvas height
c.fillStyle = "rgba(" + $('#hdn7').val() + ",0.05)";
c.fillRect(xVal, 0, width, canvas.height);
}
else if (secondDeriv > 0) {
//if concave up
//draws rectangle of width 1 and canvas height
c.fillStyle = "rgba(" + $('#hdn6').val() + ",0.05)";
c.fillRect(xVal, 0, width, canvas.height);
}
}
//if different signs
else if ((secondDeriv < 0 && previous > 0) || (secondDeriv > 0 && previous < 0)) { //|| (secondDeriv == 0)) {
//at inflection point, should draw circle
//console.log(secondDeriv);
var yVal = (-(evalExpr(i) - yMax) * canvas.height) / (yMax - yMin);
c.beginPath();
c.arc(xVal, yVal, 5, 0, 2 * Math.PI, false); //draws circle of radius centered at (xVal, yVal)
c.fillStyle = "rgba(" + $('#hdn4').val() + ",0.5)";
c.fill();
}
var previous = secondDeriv;
}
}
//var numerator;
//function to solve for when the denominator = 0;
//assumes expression to work with is assigned to expr, sets expr == denominator of function
function calculateAsymptotes() {
var denominator;
var original = expr;
for (var i = 0; i < expr.length; i++) {
if (expr.charAt(i) == '/') {
var numerator = expr.substring(0, i);
console.log(numerator);
denominator = expr.substring(i + 1);
break;
}
}
setExpr(denominator);
calculateZero(-20, 0.5, 40);
setExpr(original);
for (var j = 0; j < zeroes.length; j++) { //draws a vertical line for every place where function is undefined
//if is removable discontinuity
if (removable(zeroes[j])) {
var currentPt = (-xMin + zeroes[j]) * canvas.width / (xMax - xMin); //find x coordinate
var currentY = ((yMax - evalExpr(zeroes[j] + 0.00000001)) * canvas.height) / (yMax - yMin); //find y coordinate
c.beginPath();
c.arc(currentPt, currentY, 3, 0, 2 * Math.PI, false);
c.stroke();
} else {
var currentPt = (xMax + zeroes[j]) * canvas.width / (xMax - xMin); //find xVal
c.beginPath();
c.moveTo(currentPt, 0);
c.lineTo(currentPt, canvas.height);
c.stroke();
}
}
var numeratorDegree = 0;
var denominatorDegree = 0;
var currentPower;
for (var i = 0; i < numerator.length; i++) {
console.log(numerator.length, i, numerator.charAt(i));
if (numerator.charAt(i) == "x") {
if (numerator.charAt(i + 1) == "^") { //if currently on x and is raised to a power
currentPower = numerator.charAt(i + 2); //record power if higher than previous
} else {
currentPower = 1;
}
if (currentPower > numeratorDegree) {
numeratorDegree = currentPower;
}
}
}
for (var i = 0; i < denominator.length; i++) {
if (denominator.charAt(i) == 'x') {
if (denominator.charAt(i + 1) == '^') { //if currently on x and is raised to a power
currentPower = denominator.charAt(i + 2); //record power if higher than previous
} else {
currentPower = 1;
}
if (currentPower > denominatorDegree) {
denominatorDegree = currentPower;
}
}
}
console.log(denominatorDegree, numeratorDegree);
//calculates where to draw horizontal asymptote
if (denominatorDegree > numeratorDegree) {
//draws horizontal asymptote at y = 0 if numerator and denominator have same degree
c.beginPath();
c.moveTo(0, canvas.height / 2);
c.lineTo(canvas.width, canvas.height / 2);
c.stroke();
} else if (denominatorDegree == numeratorDegree) {
setExpr(original);
var leftAppr = evalExpr(-500);
var rightAppr = evalExpr(500);
var asymptote = (leftAppr + rightAppr) / 2;
var yVal = (-yMin + asymptote) * canvas.height / (yMax - yMin); //mapped y
yVal = 600 - yVal; //flips it to match canvas coordinates
//console.log(yVal);
c.beginPath();
c.moveTo(0, yVal);
c.lineTo(canvas.width, yVal);
c.stroke();
}
}
//precalculates symbolic derivatives and displays them in divs
//to be used later, not currently called
function displayDerivative() {
var input = $('#inputField').val(); //input from the inputfield
var firstDeriv = nerdamer('diff(' + input + ',x)').evaluate();// nerdamer(derivative of input with respect to x);
//evaluates first derivative result, sets it to div
var result = firstDeriv.evaluate();
$('#derivResult').text(result.text());
// var f = result.buildFunction();
//eval second derivative and set it to div by differentiating first derivative
var secondDeriv = nerdamer('diff(' + firstDeriv + ',x)').evaluate();
var result2 = secondDeriv.evaluate();
$('#deriv2Result').text(result2.text());
// var g = result2.buildFunction();
}
//calculates if discontinuity at xVal is removable or not, returns boolean
function removable(xVal) {
var diff1 = Math.abs(evalExpr(xVal - 0.1) - evalExpr(xVal - 0.01));
var diff2 = Math.abs(evalExpr(xVal - 0.01) - evalExpr(xVal - 0.001));
var diff3 = Math.abs(evalExpr(xVal + 0.1) - evalExpr(xVal + 0.01));
var diff4 = Math.abs(evalExpr(xVal + 0.01) - evalExpr(xVal + 0.001));
if (diff1 > diff2 && diff3 > diff4) {
return true;
}
else return false;
}
var iterations = 0; //for the function newtonZero
//recursive function, applies newton's method taking value for previous guess
//NO LONGER USED
function newtonZero(prevGuess) {
var nextGuess; //x1 in newton's
var derivative = calculateDerivative(prevGuess);
var x; //rounded x value
var y; //rounded y value
nextGuess = prevGuess - evalExpr(prevGuess) / derivative; //using the formula for newton's method
var fvalGuess = evalExpr(prevGuess);//evalExpr(nextGuess);
y = Math.round(fvalGuess * 100000) / 100000; //rounding y val to 6th place
if (Math.abs(y) < 0.00000001) { //when y val is really, really close to 0
x = Math.round(nextGuess * 100000) / 100000; //rounds to fourth decimal place. Good enough for my purposes.
return x;
}
else if (iterations < 20000 && Math.abs(y) > 0.00000001) { //if function val is not close and
//if is less than 20000th iteration
++iterations;
return (newtonZero(nextGuess)); //repeats newton's method until approximation is close enough
}
else { //if hit 20000 iterations but is still not near zero, return value //wait but what if it hasn't converged yet?
x = Math.round(nextGuess * 10000) / 10000;
return x; //returns solution or whatever value arrived at after set number of iterations.
}
}
var zeroes = [];
//a function to calculate zeroes by checking if sign has changed
function calculateZero(prevGuess, step, max) {
for (var i = step; i <= max; i += step) {
var prevX = prevGuess + i - step;
var nextX = prevGuess + i;
//calculates previous y value
var prevVal = evalExpr(prevX);
//calculates next y value
var nextVal = evalExpr(nextX);
if (nextVal == 0) {
zeroes.push(nextX);
}
else if (prevVal == 0) {
//do nothing as sign can't have changed
}
//if same sign
else if ((nextVal < 0 && prevVal < 0) || (nextVal > 0 && prevVal > 0)) {
//do nothing
}
//if different signs
else {
//add to zeroes array and continue
zeroes.push(prevGuess + i);
}
}
}
//calculates all zeroes and assigns to zeroes[];
//NOT USED
function allZeroes() {
//obsolete way, using newton's method
var nextZero;
var current = 1;
for (var i = xMin; i <= xMax ; i = i + 0.5) {
nextZero = newtonZero(i);
//if a previous zero has been calculated and the difference between them is large enough
if (zeroes[current - 1] != null && Math.abs(zeroes[current - 1]) - Math.abs(nextZero) > 0.005) {
zeroes.push(nextZero); //adds the root to array of zeroes
++current;
}
//or if is first zero calculated and has a value
else if (zeroes[current - 1] == null && !(nextZero.isNaN)) {
zeroes.push(nextZero);
++current;
}
//current keeps track of which cell of array we're on
}
//method using calculateZero()
}
//when button is clicked, graphs curves
$('#btnGraph').click(function () {
// displayDerivative();
//call function to calculate derivatives
setExpr($('#inputField').val()); //graphs main function
//setExpr($('#derivResult').text()); //graphs second derivative
if ($('#rational').is(':checked')) {
c.clearRect(0, 0, canvas.width, canvas.height);
n = 1000;
drawCurve('#' + $('#hdnFuncColor').val(), 1); //'#ff0f00', false);
generatePlane(); //draws plane again on top of function
calculateAsymptotes(); //calculates asymptotes of rational function
}
else if ($('#polynomial').is(':checked')) {
c.clearRect(0, 0, canvas.width, canvas.height);
n = 1000;
extrema(); //highlights extrema
concavity(); //marks concavity + inflection points
drawCurve('#' + $('#hdnFuncColor').val(), 1); //'#ff0f00', false);
generatePlane(); //draws plane again on top of function
drawCurve('#' + $('#hdn2').val(), 2); //graphs first derivative
drawCurve('#' + $('#hdn3').val(), 3); //graphs second derivative
}
else if ($('#other').is(':checked')) {
c.clearRect(0, 0, canvas.width, canvas.height);
n = 500;
c.clearRect(0, 0, canvas.width, canvas.height);
extrema(); //highlights extrema
concavity(); //marks concavity + inflection points
drawCurve('#' + $('#hdnFuncColor').val(), 1); //'#ff0f00', false);
generatePlane(); //draws plane again on top of function
drawCurve('#' + $('#hdn2').val(), 2); //graphs first derivative
drawCurve('#' + $('#hdn3').val(), 3); //graphs second derivative
}
});
/*$('#btnClear').click(function () {
c.clearRect(0, 0, canvas.width, canvas.height);
generatePlane();
$('#btnGraph').click();
});
*/
$('.modal-trigger').leanModal();
});