椭圆和圆之间的碰撞检测

Question

我想在椭圆和圆之间进行碰撞检测。我这样做的方式是：

1. 计算椭圆中心到圆的角度
2. 计算点在该角度的椭圆中的位置
3. 检查与该点的碰撞

但是，我遇到了一个小问题。当我计算角度时，它似乎偏离了 90 度。我做了一个肮脏的修复，简单地添加 1.5 弧度来解释 90 度，它有点工作，但存在不一致并且在某些角度不能正常工作，特别是在 0.7 和 -2.6 弧度左右。下面是代码（所有的冲突都在Ellipse类的collision方法中）

import pygame
from math import sin, cos, atan2, radians
pygame.init()

SW = 1200
SH = 600
WIN = pygame.display
D = WIN.set_mode((SW, SH))

class Circle:
    def __init__(self, radius):
        self.x = 0
        self.y = 0
        self.radius = radius

    def update(self, pos):
        self.x = pos[0]
        self.y = pos[1]

    def draw(self, display):
        pygame.draw.circle(display, (255, 0, 0), (int(self.x), int(self.y)), self.radius, 2)

circle = Circle(30)

class Ellipse:
    def __init__(self, centre, rx, ry):
        self.centre = centre
        self.collided = False
        self.rx = rx
        self.ry = ry
        
    def draw(self, display):
        angle = 0
        while angle < 6.28:
            angle += 0.001
            x = self.centre[0] + sin(angle)* self.rx
            y = self.centre[1] + cos(angle)* self.ry
            if self.collided:
                display.set_at((int(x), int(y)), (255, 0, 0))
            else:
                display.set_at((int(x), int(y)), (0, 0, 255))
        pygame.draw.circle(D, (0, 255, 0), (int(self.centre[0]), int(self.centre[1])), 5)

    def collision(self, circle):
        #angle to the circle
        dx = circle.x - self.centre[0]
        dy = circle.y - self.centre[1]
        angle = atan2(-dy, dx)
        print(angle)

        #where the point lies in the ellipse at that angle
        x = sin(angle + 1.5)* self.rx + self.centre[0] 
        y = cos(angle + 1.5)* self.ry + self.centre[1]
        #print(x, y)

        #drawing the point just to make sure its working
        # (debugging)
        pygame.draw.circle(D, (0, 255, 0), (int(x), int(y)), 5)

        # distance between the point we just
        # calculated and the circle's centre
        distance = ((x-circle.x)**2 + (y-circle.y)**2)**0.5
        #print(distance)

        #collision condition
        if distance < circle.radius:
            self.collided = True
        else:
            self.collided = False
        
ellipse = Ellipse([600, 300], 300, 200)

while True:
    events = pygame.event.get()
    mousePos = pygame.mouse.get_pos()
    for event in events:
        if event.type == pygame.QUIT:
            pygame.quit()
    D.fill((255, 255, 255))

    circle.update(mousePos)
    circle.draw(D)
    ellipse.draw(D)
    ellipse.collision(circle)
    
    pygame.display.flip()

Answer 1

第一个错误是，1.5不等于pi/2：

from math import pi

x = sin(angle + 1.5)* self.rx + self.centre[0]
y = cos(angle + 1.5)* self.ry + self.centre[1]

x = sin(angle + pi/2)* self.rx + self.centre[0] 
y = cos(angle + pi/2)* self.ry + self.centre[1]

---

其次，用角度计算椭圆上的点有点复杂。查看How to get a point on an ellipse's outline given an angle?和Calculating a Point that lies on an Ellipse given an Angle问题的答案：

from math import pi, sin, cos, atan2, radians, copysign, sqrt

class Ellipse:
    # [...]

    def pointFromAngle(self, a):
        c = cos(a)
        s = sin(a)
        ta = s / c  ## tan(a)
        tt = ta * self.rx / self.ry  ## tan(t)
        d = 1. / sqrt(1. + tt * tt)
        x = self.centre[0] + copysign(self.rx * d, c)
        y = self.centre[1] - copysign(self.ry * tt * d, s)
        return x, y

    def collision(self, circle):
        # [...]

        #where the point lies in the ellipse at that angle
        x, y = self.pointFromAngle(angle)

        # [...]

---

最小示例： repl.it/@Rabbid76/PyGame-IntersectCircleEllipse

import math
import pygame

class Circle:
    def __init__(self, center_x, center_y, radius):
        self.center = center_x, center_y
        self.radius = radius
    def update(self, center_x, center_y):
        self.center = center_x, center_y
    def draw(self, surface):
        pygame.draw.circle(surface, (255, 0, 0), (round(self.center[0]), round(self.center[1])), self.radius, 3)

class Ellipse:
    def __init__(self, center, vertex):
        self.center = center
        self.collided = False
        self.vertex = vertex
    def draw(self, surface):
        bounding_rect = pygame.Rect(0, 0, self.vertex[0] * 2, self.vertex[1] * 2)
        bounding_rect.center = round(self.center[0]), round(self.center[1])
        pygame.draw.ellipse(surface, (0, 255, 0), bounding_rect, 3)
    def pointFromAngle(self, a):
        c = math.cos(a)
        s = math.sin(a)
        ta = s / c  ## tan(a)
        tt = ta * self.vertex[0] / self.vertex[1]  ## tan(t)
        d = 1. / math.sqrt(1. + tt * tt)
        x = self.center[0] + math.copysign(self.vertex[0] * d, c)
        y = self.center[1] - math.copysign(self.vertex[1] * tt * d, s)
        return x, y

def intersect_circle_ellipse(circle, ellipse):
    dx = circle.center[0] - ellipse.center[0]
    dy = circle.center[1] - ellipse.center[1]
    angle = math.atan2(-dy, dx)
    x, y = ellipse.pointFromAngle(angle)
    distance = math.hypot(x - circle.center[0], y-circle.center[1])
    return distance <= circle.radius, (x, y) 

pygame.init()
window = pygame.display.set_mode((500, 300))
circle = Circle(0, 0, 30)      
ellipse = Ellipse(window.get_rect().center, (150, 100))
run = True
while run:
    events = pygame.event.get()
    mousePos = pygame.mouse.get_pos()
    for event in events:
        if event.type == pygame.QUIT:
            run = False
            
    circle.update(*mousePos)
    isect = intersect_circle_ellipse(circle, ellipse)

    window.fill((255, 255, 255))
    circle.draw(window)
    ellipse.draw(window) 
    color = (255, 0, 255) if isect[0] else (0, 0, 255)
    pygame.draw.circle(window, color, (round(isect[1][0]), round(isect[1][1])), 5)
    pygame.display.flip()

pygame.quit()
exit()