3D C/C++ tutorials - Resources - OpenGL perspective projection
3D C/C++ tutorials -> Resources -> OpenGL perspective projection
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To compile and run these tutorials some or all of these libraries are required: FreeImage 3.16.0, GLEW 1.11.0, GLUT 3.7.6 / GLUT for Dev-C++, GLM 0.9.5.4
mat4x4 M - model matrix - model-to-world space transformation, the result of several matrices multiplication (translation, rotation, scaling) may be stored in it
mat4x4 V - view matrix - world-to-view space transformation
mat4x4 P - perspective projection matrix - view-to-clip and clip-to-ndc space transformation (perspective division must be applied)
mat4x4 Bias - bias matrix - ndc-to-texture space transformation (shadow mapping, screen space ambient occlusion, ...)
mat4x4 Vp - viewport matrix - viewport transformation

vec4 vm = vec4(x, y, z, 1) - vertex in model space

vec4 vw = M * vm - vertex in world space

vec4 vv = V * vw - vertex in view (camera, eye) space

V causes vw to be rotated and translated in the opposite direction considering the rotation and position of the camera

if vv.z < 0 then vv is in front of the camera

vec4 vc = P * vv - vertex in clip space

vc.x = P[0] * vv.x
vc.y = P[5] * vv.y
vc.z = P[10] * vv.z + P[14] * vv.w = P[10] * vv.z + P[14]
vc.w = P[11] * vv.z = -vv.z

vc = P * V * M * vm = (P * (V * (M * vm))) - output of a vertex shader - gl_Position

For performance reasons the result of the model and view matrix multiplication is stored in the modelview matrix (GL_MODELVIEW_MATRIX, gl_ModelViewMatrix) MV = V * M; and the result of the modelview and projection matrix multiplication is stored in the modelviewprojection matrix (gl_ModelViewProjectionMatrix) MVP = P * MV = P * V * M;.

if vc.w > 0 then vc is in front of the camera

if (-vc.w <= vc.x < vc.w) and (-vc.w <= vc.y < vc.w) and (-vc.w <= vc.z < vc.w) then vc is inside the camera frustum and is visible on the screen

vc.w != 1 therefore vc must be divided by vc.w

vndc = vc / vc.w - perspective division

if (-1 <= vndc.x < 1) and (-1 <= vndc.y < 1) and (-1 <= vndc.z < 1) then vndc is inside the camera frustum and is visible on the screen

vec4 vt = Bias * vndc - vertex in texture space

vt.x = vndc.x * 0.5 + 0.5
vt.y = vndc.y * 0.5 + 0.5
vt.z = vndc.z * 0.5 + 0.5

vt.x and vt.y are used to fetch the depth texture and vt.z is used for depth comparison (shadow mapping, screen space ambient occlusion, ...)

if (0 <= vt.x < 1) and (0 <= vt.y < 1) and (0 <= vt.z < 1) then vt is inside the camera frustum

and if vt lies in the

  • near clip plane then vt.z = 0
  • far clip plane then vt.z = 1

vt.z is used for depth test and stored in the depth buffer

vec4 p = Vp * vt - pixel on the screen

p.x = vt.x * width + x
p.y = vt.y * height + y

p.x and p.y are coordinates to the color and depth buffers

if a vertex vt is inside the camera frustum then

x <= p.x < x + width
y <= p.y < y + height

See the Simple software renderer or Software shadow mapping tutorials for more details.


View matrix - gluLookAt(ex, ey, ez, cx, cy, cz, ux, uy, uz)

vec3 Pc = vec3(ex, ey, ez);

vec3 Zc = normalize(Pc - vec3(cx, cy, cz));

vec3 Yc = normalize(vec3(ux, uy, uz));

vec3 Xc = normalize(cross(Yc, Zc));

Yc = cross(Zc, Xc);



Perspective projection matrix - gluPerspective(fovy, aspect, n, f)



Bias matrix



Viewport matrix - glViewport(x, y, width, height)

© 2010 - 2016 Bc. Michal Belanec, michalbelanec@centrum.sk
Last update June 25, 2016
OpenGL® is a registered trademark of Silicon Graphics Inc.