球-三角形碰撞
进球：我有一个三角形的球。球有一个初始位置和速度。我在试着弄清楚球会打到三角形的哪一边。
我尝试过的方法：I derived a formula通过将球的路径和三角形的边参数化，并找到满足参数方程的最短时间，输出球将击中哪一边。但是当我将这个公式应用到我的程序中时，它产生了错误的结果！我试了很多方法，但都没有用。如有任何帮助，我们不胜感激。MWE在此：CodePen
数据-lang="js"数据-隐藏="真"数据-控制台="真"数据-巴贝尔="假">let angle = 0;
let sides = [];
let vertices = [];
const len = 100;
function setup() {
createCanvas(windowWidth, windowHeight);
angleMode(DEGREES);
angleOne = createSlider(0, 89, 60);
angleOne.position(10, 10);
angleOne.style("width", "80px");
angleTwo = createSlider(0, 89, 60);
angleTwo.position(10, 30);
angleTwo.style("width", "80px");
// Initial vertice & side setup (these don't change)
let v1 = createVector(width / 2 - len / 2, height * 0.7);
let v2 = createVector(width / 2 + len / 2, height * 0.7);
sides[0] = new Side(v1.x, v1.y, v2.x, v2.y, "green");
vertices[0] = new Vertex(v1.x, v1.y);
vertices[1] = new Vertex(v2.x, v2.y);
}
function draw() {
background(255);
let angOne = angleOne.value();
let angTwo = angleTwo.value();
fill(0);
strokeWeight(0);
textSize(15);
text(angOne, 100, 25);
text(angTwo, 100, 45);
let v2Offset = createVector(len * cos(-angOne), len * sin(-angOne));
let v3Offset = createVector(-len * cos(angTwo), -len * sin(angTwo));
vertices[2] = new Vertex(
vertices[0].a.x + v2Offset.x,
vertices[0].a.y + v2Offset.y
);
vertices[3] = new Vertex(
vertices[1].a.x + v3Offset.x,
vertices[1].a.y + v3Offset.y
);
// Update the sides
sides[1] = new Side(
vertices[0].a.x,
vertices[0].a.y,
vertices[2].a.x,
vertices[2].a.y
);
sides[3] = new Side(
vertices[1].a.x,
vertices[1].a.y,
vertices[3].a.x,
vertices[3].a.y
);
const m1 =
(vertices[2].a.y - vertices[0].a.y) / (vertices[2].a.x - vertices[0].a.x);
const m2 =
(vertices[3].a.y - vertices[1].a.y) / (vertices[3].a.x - vertices[1].a.x);
// Calculate the y-offset relative to vertices[0]
const b2 = (vertices[1].a.x - vertices[0].a.x) * -m2;
const xInt = b2 / (m1 - m2);
const yInt = xInt * m1;
// Note xInt and yInt are relative to vertices[0]
// draw all the things
// sides.forEach((s) => s.show());
// stroke(0, 255, 0);
// strokeWeight(20);
point(vertices[0].a.x + xInt, vertices[0].a.y + yInt);
vertices[4] = new Vertex(vertices[0].a.x + xInt, vertices[0].a.y + yInt);
sides[4] = new Side(
vertices[1].a.x,
vertices[1].a.y,
vertices[0].a.x + xInt,
vertices[0].a.y + yInt,
"blue"
);
sides[5] = new Side(
vertices[0].a.x,
vertices[0].a.y,
vertices[0].a.x + xInt,
vertices[0].a.y + yInt,
"purple"
);
scale(2); // so I can make the triangle actually *visible*
translate(-width / 3, -height / 4);
sides[0].show();
sides[4].show();
sides[5].show();
vertices[0].show();
vertices[1].show();
vertices[4].show();
strokeWeight(1);
stroke(255, 0, 0);
noFill();
arc(vertices[0].a.x, vertices[0].a.y, 40, 40, -1 * angleOne.value(), 0, PIE);
arc(
vertices[1].a.x,
vertices[1].a.y,
40,
40, -180, -(180 - angleTwo.value()),
PIE
);
let P1x = vertices[0].a.x;
let P1y = vertices[0].a.y;
let P2x = vertices[1].a.x;
let P2y = vertices[1].a.y;
let P3x = vertices[4].a.x;
let P3y = vertices[4].a.y;
stroke(255, 255, 0);
stroke("purple"); // Change the color
strokeWeight(5); // Make the points 10 pixels in size
let P0 = createVector(P1x + 60, P1y - 40);
let V0 = createVector(0, -15);
point(P0.x, P0.y);
stroke(255, 0, 0);
point(P0.x + V0.x, P0.y + V0.y);
strokeWeight(2);
stroke("purple");
line(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
// console.log(P1x,P1y,P2x,P2y,P3x,P3y);
let A1 = P3y - P1y;
let B1 = -(P3x - P1x);
let C1 = A1 * P1x + B1 * P1y;
let A2 = -(P3y - P2y);
let B2 = P3x - P2x;
let C2 = A2 * P2x + B2 * P2y;
let A3 = -(P2y - P1y);
let B3 = P2x - P1x;
let C3 = A3 * P2x + B3 * P2y;
let t1 = (C1 - A1 * P0.x - B1 * P0.y) / (A1 * V0.x + B1 * P0.y);
let t2 = (C2 - A2 * P0.x - B2 * P0.y) / (A2 * V0.x + B2 * P0.y);
let t3 = (C3 - A3 * P0.x - B3 * P0.y) / (A3 * V0.x + B3 * P0.y);
let times = [t1, t2, t3];
let posTimes = [];
for (let i = 0; i < times.length; i++) {
times[i] = round(times[i], 2);
}
// console.log("After rounding:", times);
for (let i = 0; i < times.length; i++) {
if (times[i] > 0) {
posTimes.push(times[i]);
}
}
// console.log("posTimes:", posTimes);
trueTime = min(posTimes);
if (trueTime == round(t1, 2)) {
fill("Blue");
text("Hit Blue", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
} else if (trueTime == round(t2, 2)) {
fill("Green");
text("Hit Green", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
} else {
fill("Purple");
text("Hit Purple", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
}
}
class Side {
constructor(x1, y1, x2, y2, col = "black") {
this.a = createVector(x1, y1);
this.b = createVector(x2, y2);
this.color = col;
}
show() {
stroke(this.color);
strokeWeight(4);
line(this.a.x, this.a.y, this.b.x, this.b.y);
}
}
class Vertex {
constructor(x1, y1) {
this.a = createVector(x1, y1);
}
show() {
stroke(255, 0, 0);
strokeWeight(10);
point(this.a.x, this.a.y);
}
}html, body { margin: 0; padding: 0; overflow: hidden }<script src="https://cdn.jsdelivr.net/npm/p5@1.4.1/lib/p5.min.js"></script>解决方案
我搞不懂你的数学。我认为您应该尝试用解释性注释来注释这类代码。通常，这将帮助您发现自己的错误：
let A1 = P3y - P1y;
let B1 = -(P3x - P1x);
let C1 = A1 * P1x + B1 * P1y;
let A2 = -(P3y - P2y);
let B2 = P3x - P2x;
let C2 = A2 * P2x + B2 * P2y;
let A3 = -(P2y - P1y);
let B3 = P2x - P1x;
let C3 = A3 * P2x + B3 * P2y;
let t1 = (C1 - A1 * P0.x - B1 * P0.y) / (A1 * V0.x + B1 * P0.y);
let t2 = (C2 - A2 * P0.x - B2 * P0.y) / (A2 * V0.x + B2 * P0.y);
let t3 = (C3 - A3 * P0.x - B3 * P0.y) / (A3 * V0.x + B3 * P0.y);
这里有一些实用的数学方法。它可能不是最密集/最优雅的，但我试图从基本代数一步一步地解释它：
// Find which of
//
// P1 to P2 = green
// P1 to P3 = purple
// P2 to P3 = blue
//
// PO to PO + V0 intersects with
// Find the intersection point between lines A and B
function intersection(Ax1, Ay1, Ax2, Ay2, Bx1, By1, Bx2, By2) {
// Calculate the slope of line A
let Am = (Ay2 - Ay1) / (Ax2 - Ax1);
// Calculate the y-intercept of line A
let Ab = Ay1 - Ax1 * Am;
// slope of line B
let Bm = (By2 - By1) / (Bx2 - Bx1);
// y-intercept of line B
let Bb = By1 - Bx1 * Bm;
if (Am === Bm) {
// Parallel lines
return;
}
if (!Number.isFinite(Am)) {
// Line A is vertical
if (!Number.isFinite(Bm)) {
// Line B is also vertical (Am may not equal Bm though because Infinity != NegativeInfinity)
return;
} else {
// Since line A is vertical, intersection point will lie along the same x position as Ax1 and Ax2
const xInt = Ax1;
// Simply use the equation for line segment B to find the corresponding Y value
const yInt = Bm * xInt + Bb;
return createVector(xInt, yInt);
}
} else if (!Number.isFinite(Bm)) {
// Line B is vertical
const xInt = Bx1;
const yInt = Am * xInt + Ab;
return createVector(xInt, yInt);
} else {
// Derived from Am * x + Ab = Bm * x + Bb
const xInt = (Bb - Ab) / (Am - Bm);
const yInt = Am * xInt + Ab;
return createVector(xInt, yInt);
}
}
let P1toP2int =
intersection(P1.x, P1.y, P2.x, P2.y, P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
let P1toP3int =
intersection(P1.x, P1.y, P3.x, P3.y, P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
let P2toP3int =
intersection(P2.x, P2.y, P3.x, P3.y, P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
// These intersection points assume that all lines point infinitely in both
// directions, so we still have some more work to do.
// Check if each of these points is within the target segment
function isWithin(lineX1, lineY1, lineX2, lineY2, xInt, yInt) {
if (abs((lineY2 - lineY1) / (lineX2 - lineX1)) > 1) {
// If the line segment is more vertical, check the Y position
return yInt >= min(lineY1, lineY2) && yInt <= max(lineY1, lineY2);
} else {
return xInt >= min(lineX1, lineX2) && xInt <= max(lineX1, lineX2);
}
}
if (P1toP2int && !isWithin(P1.x, P1.y, P2.x, P2.y, P1toP2int.x, P1toP2int.y)) {
P1toP2int = undefined;
}
if (P1toP3int && !isWithin(P1.x, P1.y, P3.x, P3.y, P1toP3int.x, P1toP3int.y)) {
P1toP3int = undefined;
}
if (P2toP3int && !isWithin(P2.x, P2.y, P3.x, P3.y, P2toP3int.x, P2toP3int.y)) {
P2toP3int = undefined;
}
// Check if each intersection point is in the direction our ray is pointing
function isOnRay(rayX0, rayY0, rayX1, rayY1, xInt, yInt) {
// If the ray is more vertical, check the y coordinates
if (abs((rayY1 - rayY0) / (rayX1 - rayX0)) > 1) {
// If the ray is pointing in the positive Y direction
// (rayY1 > rayY0) then the yInt must be on the positive
// side of rayY0; and vice versa
return (rayY1 > rayY0) === (yInt > rayY0);
} else {
return (rayX1 > rayX0) === (xInt > rayX0);
}
}
if (P1toP2int && !isOnRay(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y, P1toP2int.x, P1toP2int.y)) {
P1toP2int = undefined;
}
if (P1toP3int && !isOnRay(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y, P1toP3int.x, P1toP3int.y)) {
P1toP3int = undefined;
}
if (P2toP3int && !isOnRay(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y, P2toP3int.x, P2toP3int.y)) {
P2toP3int = undefined;
}
let angle = 0;
let sides = [];
let vertices = [];
const len = 100;
let angleOne;
let angleTwo;
let angleThree;
function setup() {
createCanvas(windowWidth, windowHeight);
angleMode(DEGREES);
angleOne = createSlider(0, 89, 60);
angleOne.position(10, 10);
angleOne.style("width", "80px");
angleTwo = createSlider(0, 89, 60);
angleTwo.position(10, 30);
angleTwo.style("width", "80px");
angleThree = createSlider(0, 360, 0);
angleThree.position(10, 50);
angleThree.style("width", "80px");
// Initial vertice & side setup (these don't change)
let v1 = createVector(width / 2 - len / 2, height * 0.7);
let v2 = createVector(width / 2 + len / 2, height * 0.7);
sides[0] = new Side(v1.x, v1.y, v2.x, v2.y, "green");
vertices[0] = new Vertex(v1.x, v1.y);
vertices[1] = new Vertex(v2.x, v2.y);
}
function draw() {
background(255);
let angOne = angleOne.value();
let angTwo = angleTwo.value();
let rayAngle = angleThree.value();
fill(0);
strokeWeight(0);
textSize(15);
text(angOne, 100, 25);
text(angTwo, 100, 45);
text(rayAngle, 100, 65);
let v2Offset = createVector(len * cos(-angOne), len * sin(-angOne));
let v3Offset = createVector(-len * cos(angTwo), -len * sin(angTwo));
vertices[2] = new Vertex(
vertices[0].a.x + v2Offset.x,
vertices[0].a.y + v2Offset.y
);
vertices[3] = new Vertex(
vertices[1].a.x + v3Offset.x,
vertices[1].a.y + v3Offset.y
);
// Update the sides
sides[1] = new Side(
vertices[0].a.x,
vertices[0].a.y,
vertices[2].a.x,
vertices[2].a.y
);
sides[3] = new Side(
vertices[1].a.x,
vertices[1].a.y,
vertices[3].a.x,
vertices[3].a.y
);
const m1 =
(vertices[2].a.y - vertices[0].a.y) / (vertices[2].a.x - vertices[0].a.x);
const m2 =
(vertices[3].a.y - vertices[1].a.y) / (vertices[3].a.x - vertices[1].a.x);
// Calculate the y-offset relative to vertices[0]
const b2 = (vertices[1].a.x - vertices[0].a.x) * -m2;
const xInt = b2 / (m1 - m2);
const yInt = xInt * m1;
// Note xInt and yInt are relative to vertices[0]
// draw all the things
// sides.forEach((s) => s.show());
// stroke(0, 255, 0);
// strokeWeight(20);
point(vertices[0].a.x + xInt, vertices[0].a.y + yInt);
vertices[4] = new Vertex(vertices[0].a.x + xInt, vertices[0].a.y + yInt);
sides[4] = new Side(
vertices[1].a.x,
vertices[1].a.y,
vertices[4].a.x,
vertices[4].a.y,
"blue"
);
sides[5] = new Side(
vertices[0].a.x,
vertices[0].a.y,
vertices[4].a.x,
vertices[4].a.y,
"purple"
);
scale(2); // so I can make the triangle actually *visible*
translate(-width / 3, -height / 4);
sides[0].show();
sides[4].show();
sides[5].show();
vertices[0].show();
vertices[1].show();
vertices[4].show();
strokeWeight(1);
stroke(255, 0, 0);
noFill();
arc(vertices[0].a.x, vertices[0].a.y, 40, 40, -1 * angleOne.value(), 0, PIE);
arc(
vertices[1].a.x,
vertices[1].a.y,
40,
40, -180, -(180 - angleTwo.value()),
PIE
);
let P1 = vertices[0].a;
let P2 = vertices[1].a;
let P3 = vertices[4].a;
stroke(255, 255, 0);
stroke("purple"); // Change the color
strokeWeight(5); // Make the points 10 pixels in size
let P0 = createVector(P1.x + 60, P1.y - 40);
let V0 = createVector(15 * cos(rayAngle), 15 * sin(rayAngle));
point(P0.x, P0.y);
stroke(255, 0, 0);
point(P0.x + V0.x, P0.y + V0.y);
strokeWeight(2);
stroke("purple");
line(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
// console.log(P1x,P1y,P2x,P2y,P3x,P3y);
// Find which of
//
// P1 to P2 = green
// P1 to P3 = purple
// P2 to P3 = blue
//
// PO to PO + V0 intersects with
// Find the intersection point between lines A and B
function intersection(Ax1, Ay1, Ax2, Ay2, Bx1, By1, Bx2, By2) {
// Calculate the slope of line A
let Am = (Ay2 - Ay1) / (Ax2 - Ax1);
// Calculate the y-intercept of line A
let Ab = Ay1 - Ax1 * Am;
// slope of line B
let Bm = (By2 - By1) / (Bx2 - Bx1);
// y-intercept of line B
let Bb = By1 - Bx1 * Bm;
if (Am === Bm) {
// Parallel lines
return;
}
if (!Number.isFinite(Am)) {
// Line A is vertical
if (!Number.isFinite(Bm)) {
// Line B is also vertical (Am may not equal Bm though because Infinity != NegativeInfinity)
return;
} else {
// Since line A is vertical, intersection point will lie along the same x position as Ax1 and Ax2
const xInt = Ax1;
// Simply use the equation for line segment B to find the corresponding Y value
const yInt = Bm * xInt + Bb;
return createVector(xInt, yInt);
}
} else if (!Number.isFinite(Bm)) {
// Line B is vertical
const xInt = Bx1;
const yInt = Am * xInt + Ab;
return createVector(xInt, yInt);
} else {
// Derived from Am * x + Ab = Bm * x + Bb
const xInt = (Bb - Ab) / (Am - Bm);
const yInt = Am * xInt + Ab;
return createVector(xInt, yInt);
}
}
let P1toP2int =
intersection(P1.x, P1.y, P2.x, P2.y, P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
let P1toP3int =
intersection(P1.x, P1.y, P3.x, P3.y, P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
let P2toP3int =
intersection(P2.x, P2.y, P3.x, P3.y, P0.x, P0.y, P0.x + V0.x, P0.y + V0.y);
// These intersection points assume that all lines point infinitely in both
// directions, so we still have some more work to do.
// Check if each of these points is within the target segment
function isWithin(lineX1, lineY1, lineX2, lineY2, xInt, yInt) {
if (abs((lineY2 - lineY1) / (lineX2 - lineX1)) > 1) {
// If the line segment is more vertical, check the Y position
return yInt >= min(lineY1, lineY2) && yInt <= max(lineY1, lineY2);
} else {
return xInt >= min(lineX1, lineX2) && xInt <= max(lineX1, lineX2);
}
}
if (P1toP2int && !isWithin(P1.x, P1.y, P2.x, P2.y, P1toP2int.x, P1toP2int.y)) {
P1toP2int = undefined;
}
if (P1toP3int && !isWithin(P1.x, P1.y, P3.x, P3.y, P1toP3int.x, P1toP3int.y)) {
P1toP3int = undefined;
}
if (P2toP3int && !isWithin(P2.x, P2.y, P3.x, P3.y, P2toP3int.x, P2toP3int.y)) {
P2toP3int = undefined;
}
// Check if each intersection point is in the direction our ray is pointing
function isOnRay(rayX0, rayY0, rayX1, rayY1, xInt, yInt) {
// If the ray is more vertical, check the y coordinates
if (abs((rayY1 - rayY0) / (rayX1 - rayX0)) > 1) {
// If the ray is pointing in the positive Y direction
// (rayY1 > rayY0) then the yInt must be on the positive
// side of rayY0; and vice versa
return (rayY1 > rayY0) === (yInt > rayY0);
} else {
return (rayX1 > rayX0) === (xInt > rayX0);
}
}
if (P1toP2int && !isOnRay(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y, P1toP2int.x, P1toP2int.y)) {
P1toP2int = undefined;
}
if (P1toP3int && !isOnRay(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y, P1toP3int.x, P1toP3int.y)) {
P1toP3int = undefined;
}
if (P2toP3int && !isOnRay(P0.x, P0.y, P0.x + V0.x, P0.y + V0.y, P2toP3int.x, P2toP3int.y)) {
P2toP3int = undefined;
}
// Only one of these should be true, except perhaps if the ray passes precisely through a corner
if (P1toP2int) {
stroke("Red");
strokeWeight(8);
point(P1toP2int.x, P1toP2int.y);
fill("Green");
noStroke();
text("Hit Green", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
}
if (P1toP3int) {
stroke("Red");
strokeWeight(8);
point(P1toP3int.x, P1toP3int.y);
fill("Purple");
noStroke();
text("Hit Purple", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
}
if (P2toP3int) {
stroke("Red");
strokeWeight(8);
point(P2toP3int.x, P2toP3int.y);
fill("Blue");
noStroke();
text("Hit Blue", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
}
/* I don't understand this math at all
let A1 = P3y - P1y;
let B1 = -(P3x - P1x);
let C1 = A1 * P1x + B1 * P1y;
let A2 = -(P3y - P2y);
let B2 = P3x - P2x;
let C2 = A2 * P2x + B2 * P2y;
let A3 = -(P2y - P1y);
let B3 = P2x - P1x;
let C3 = A3 * P2x + B3 * P2y;
let t1 = (C1 - A1 * P0.x - B1 * P0.y) / (A1 * V0.x + B1 * P0.y);
let t2 = (C2 - A2 * P0.x - B2 * P0.y) / (A2 * V0.x + B2 * P0.y);
let t3 = (C3 - A3 * P0.x - B3 * P0.y) / (A3 * V0.x + B3 * P0.y);
let times = [t1, t2, t3];
let posTimes = [];
for (let i = 0; i < times.length; i++) {
times[i] = round(times[i], 2);
}
// console.log("After rounding:", times);
for (let i = 0; i < times.length; i++) {
if (times[i] > 0) {
posTimes.push(times[i]);
}
}
// console.log("posTimes:", posTimes);
trueTime = min(posTimes);
if (trueTime == round(t1, 2)) {
fill("Blue");
text("Hit Blue", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
} else if (trueTime == round(t2, 2)) {
fill("Green");
text("Hit Green", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
} else {
fill("Purple");
text("Hit Purple", vertices[1].a.x + 50, max(50, vertices[1].a.y - 50));
}
*/
}
class Side {
constructor(x1, y1, x2, y2, col = "black") {
this.a = createVector(x1, y1);
this.b = createVector(x2, y2);
this.color = col;
}
show() {
stroke(this.color);
strokeWeight(4);
line(this.a.x, this.a.y, this.b.x, this.b.y);
}
}
class Vertex {
constructor(x1, y1) {
this.a = createVector(x1, y1);
}
show() {
stroke(255, 0, 0);
strokeWeight(10);
point(this.a.x, this.a.y);
}
}html, body { margin: 0; padding: 0; overflow: hidden }<script src="https://cdn.jsdelivr.net/npm/p5@1.4.1/lib/p5.min.js"></script>相关文章