import numpy as np def vec3(x, y, z): return np.array([x, y, z], dtype=np.float32) def radians(v): return np.radians(v) def identity(): return np.identity(4, dtype=np.float32) def empty(): return np.zeros([4, 4], dtype=np.float32) def magnitude(v): return np.linalg.norm(v) def normalize(v): m = magnitude(v) return v if m == 0 else v / m def dot(u, v): return np.sum(u * v) def cross(u, v): res = vec3(0, 0, 0) res[0] = u[1] * v[2] - u[2] * v[1] res[1] = u[2] * v[0] - u[0] * v[2] res[2] = u[0] * v[1] - u[1] * v[0] return res # below functions can be optimized def translate(m, v): res = np.copy(m) res[:, 3] = m[:, 0] * v[0] + m[:, 1] * v[1] + m[:, 2] * v[2] + m[:, 3] return res def rotate(m, angle, v): a = angle c = np.cos(a) s = np.sin(a) axis = normalize(v) temp = (1 - c) * axis rot = empty() rot[0][0] = c + temp[0] * axis[0] rot[0][1] = temp[0] * axis[1] + s * axis[2] rot[0][2] = temp[0] * axis[2] - s * axis[1] rot[1][0] = temp[1] * axis[0] - s * axis[2] rot[1][1] = c + temp[1] * axis[1] rot[1][2] = temp[1] * axis[2] + s * axis[0] rot[2][0] = temp[2] * axis[0] + s * axis[1] rot[2][1] = temp[2] * axis[1] - s * axis[0] rot[2][2] = c + temp[2] * axis[2] res = empty() res[:, 0] = m[:, 0] * rot[0][0] + m[:, 1] * rot[0][1] + m[:, 2] * rot[0][2] res[:, 1] = m[:, 0] * rot[1][0] + m[:, 1] * rot[1][1] + m[:, 2] * rot[1][2] res[:, 2] = m[:, 0] * rot[2][0] + m[:, 1] * rot[2][1] + m[:, 2] * rot[2][2] res[:, 3] = m[:, 3] return res def perspective(fovy, aspect, zNear, zFar): tanHalfFovy = np.tan(fovy / 2) res = empty() res[0][0] = 1 / (aspect * tanHalfFovy) res[1][1] = 1 / (tanHalfFovy) res[2][3] = -1 res[2][2] = - (zFar + zNear) / (zFar - zNear) res[3][2] = -(2 * zFar * zNear) / (zFar - zNear) return res.T def ortho(left, right, bottom, top, zNear, zFar): # res = np.ones([4, 4], dtype=np.float32) res = identity() res[0][0] = 2 / (right - left) res[1][1] = 2 / (top - bottom) res[2][2] = - 2 / (zFar - zNear) res[3][0] = - (right + left) / (right - left) res[3][1] = - (top + bottom) / (top - bottom) res[3][2] = - (zFar + zNear) / (zFar - zNear) return res.T def lookat(eye, center, up): f = normalize(center - eye) s = normalize(cross(f, up)) u = cross(s, f) res = identity() res[0][0] = s[0] res[1][0] = s[1] res[2][0] = s[2] res[0][1] = u[0] res[1][1] = u[1] res[2][1] = u[2] res[0][2] = -f[0] res[1][2] = -f[1] res[2][2] = -f[2] res[3][0] = -dot(s, eye) res[3][1] = -dot(u, eye) res[3][2] = -dot(f, eye) return res.T def transform(d, m): return np.dot(m, d.T).T