| 07a62564 |
#!/usr/bin/env python3
from sympy import *
def mat(*args):
return Matrix(list(map(list, args))).T
def vec(*args):
return Matrix(args)
def normalize(v):
return v / sqrt(v.dot(v))
def elementwise(a, b, f):
return Matrix([
[
f(a[r, c], b[r, c])
for c in range(a.cols)
]
for r in range(a.rows)
])
def mul(a, b):
return elementwise(a, b, lambda a, b: a * b)
def div(a, b):
return elementwise(a, b, lambda a, b: a / b)
def window(viewport, depth_range):
size = vec(*viewport[2:4], depth_range[1]-depth_range[0])
offs = vec(*viewport[0:2], depth_range[0])
s = 0.5 * size
t = 0.5 * size + offs
return Matrix([
[s[0], 0, 0, t[0]],
[0, s[1], 0, t[1]],
[0, 0, s[2], t[2]],
[0, 0, 0, 1],
])
def ortho(left, right, bottom, top, near, far):
size = vec(right-left, top-bottom, far-near)
offs = vec( -left, -bottom, +near)
s = div(vec(2, 2, -2), size)
t = mul(s, offs) - vec(1, 1, 1)
return Matrix([
[s[0], 0, 0, t[0]],
[0, s[1], 0, t[1]],
[0, 0, s[2], t[2]],
[0, 0, 0, 1],
])
def frustum(left, right, bottom, top, near, far):
z = vec(0, 0, near+far, -1)
w = vec(0, 0, near*far, 0)
return ortho(left, right, bottom, top, near, far) * Matrix([
[near, 0, z[0], z[0]],
[0, near, z[1], w[1]],
[0, 0, z[2], w[2]],
[0, 0, z[3], w[3]],
])
def perspective(fovy, aspect, near, far):
y = near * tan(0.5 * fovy)
x = y * aspect
return frustum(-x, +x, -y, +y, near, far)
def lookat(position, target, up):
z = normalize(position - target)
x = normalize(up.cross(z))
y = z.cross(x)
R_inv = mat(x, y, z).T
t_inv = -position
return mat(R_inv.row(0), R_inv.row(0), R_inv.row(0), R_inv * t_inv)
# https://registry.khronos.org/OpenGL-Refpages/gl4/html/glViewport.xhtml
# https://registry.khronos.org/OpenGL-Refpages/gl4/html/glDepthRange.xhtml
# https://www.khronos.org/opengl/wiki/Vertex_Post-Processing#Viewport_transform
# https://www.songho.ca/opengl/gl_viewport.html
viewport = symbols('x, y, width, height')
depth_range = symbols('near, far')
pprint(simplify(window(viewport, depth_range)))
# ⎡0.5⋅width 0 0 0.5⋅width + x ⎤
# ⎢ ⎥
# ⎢ 0 0.5⋅height 0 0.5⋅height + y ⎥
# ⎢ ⎥
# ⎢ 0 0 0.5⋅far - 0.5⋅near 0.5⋅far + 0.5⋅near⎥
# ⎢ ⎥
# ⎣ 0 0 0 1 ⎦
# https://registry.khronos.org/OpenGL-Refpages/gl2.1/xhtml/glOrtho.xml
# https://www.songho.ca/opengl/gl_projectionmatrix.html#ortho
left, right, bottom, top, near, far = symbols('left, right, bottom, top, near, far')
pprint(simplify(ortho(left, right, bottom, top, near, far)))
# ⎡ -2 left + right⎤
# ⎢──────────── 0 0 ────────────⎥
# ⎢left - right left - right⎥
# ⎢ ⎥
# ⎢ -2 bottom + top⎥
# ⎢ 0 ──────────── 0 ────────────⎥
# ⎢ bottom - top bottom - top⎥
# ⎢ ⎥
# ⎢ -2 -far - near ⎥
# ⎢ 0 0 ────────── ─────────── ⎥
# ⎢ far - near far - near ⎥
# ⎢ ⎥
# ⎣ 0 0 0 1 ⎦
# https://registry.khronos.org/OpenGL-Refpages/gl2.1/xhtml/glFrustum.xml
# https://www.songho.ca/opengl/gl_projectionmatrix.html#perspective
left, right, bottom, top, near, far = symbols('left, right, bottom, top, near, far')
pprint(simplify(frustum(left, right, bottom, top, near, far)))
# ⎡ -2⋅near -left - right ⎤
# ⎢──────────── 0 ───────────── 0 ⎥
# ⎢left - right left - right ⎥
# ⎢ ⎥
# ⎢ -2⋅near -bottom - top ⎥
# ⎢ 0 ──────────── ───────────── 0 ⎥
# ⎢ bottom - top bottom - top ⎥
# ⎢ ⎥
# ⎢ -far - near -2⋅far⋅near ⎥
# ⎢ 0 0 ─────────── ────────────⎥
# ⎢ far - near far - near ⎥
# ⎢ ⎥
# ⎣ 0 0 -1 0 ⎦
# https://registry.khronos.org/OpenGL-Refpages/gl2.1/xhtml/gluPerspective.xml
# https://www.songho.ca/opengl/gl_projectionmatrix.html#fov
fovy, aspect, near, far = symbols('fovy, aspect, near, far')
pprint(simplify(perspective(fovy, aspect, near, far)))
# ⎡ 1 ⎤
# ⎢──────────────────── 0 0 0 ⎥
# ⎢aspect⋅tan(0.5⋅fovy) ⎥
# ⎢ ⎥
# ⎢ 1 ⎥
# ⎢ 0 ───────────── 0 0 ⎥
# ⎢ tan(0.5⋅fovy) ⎥
# ⎢ ⎥
# ⎢ -far - near -2⋅far⋅near ⎥
# ⎢ 0 0 ─────────── ────────────⎥
# ⎢ far - near far - near ⎥
# ⎢ ⎥
# ⎣ 0 0 -1 0 ⎦
# https://registry.khronos.org/OpenGL-Refpages/gl2.1/xhtml/gluLookAt.xml
# https://www.songho.ca/opengl/gl_camera.html#lookat
position = Matrix(MatrixSymbol('position', 3, 1))
target = Matrix(MatrixSymbol('target', 3, 1))
up = Matrix(MatrixSymbol('up', 3, 1))
# pprint(simplify(lookat(position, target, up)))
|
git.rcrnstn.net