viewing and projection
DESCRIPTION
Viewing and Projection. The topics. Interior parameters Projection type Field of view Clipping Frustum… Exterior parameters Camera position Camera orientation. Transformation Pipeline. Local coordinate. Local->World. ModelView Matrix. World coordinate. World->Eye. Eye coordinate. - PowerPoint PPT PresentationTRANSCRIPT
Viewing and Projection
The topics
• Interior parameters• Projection type• Field of view• Clipping• Frustum…
• Exterior parameters• Camera position• Camera orientation
World coordinate
others
Transformation Pipeline
Projection Matrix
World->Eye
Local coordinate
Eye coordinate
Local->World
ModelView Matrix
Clip coordinate
Screen coordinate
Projection
• The projection transforms a point from a high-dimensional space to a low-dimensional space.
• In 3D, the projection means mapping a 3D point onto a 2D projection plane (or called image plane).
• There are two basic projection types: • Parallel: orthographic, oblique• Perspective
Orthographic ProjectionImage Plane
Direction of Projection
xyk1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
1 0 0 00 1 0 00 0 0 k0 0 0 1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
z=k
z-axis
Orthographic Projection
Oblique Projection
Image Plane
Direction of Projection
Properties of Parallel Projection
• Definition: projection directions are parallel.
• Doesn’t look real.
• Can preserve parallel lines
Projection
Parallel in 3D Parallel in 2D
Properties of Parallel Projection
• Definition: projection directions are parallel.
• Doesn’t look real.
• Can preserve parallel lines• Can preserve ratios
Projections
t
s '
t '
s : t =s':t'
Properties of Parallel Projection
• Definition: projection directions are parallel.
• Doesn’t look real.
• Can preserve parallel lines• Can preserve ratios• CANNOT preserve angles
Projection
Properties of Parallel Projection
• Definition: projection directions are parallel.
• Doesn’t look real.
• Can preserve parallel lines• Can preserve ratios• CANNOT preserve angles
• Often used in CAD, architecture drawings, when images can be used for measurement.
Properties of Parallel Projection
• No foreshortening
Image Plane
Perspective Projection
• Perspective projection has foreshortening:
Image Plane
Center of Projection
Perspective Projection• Images are mapped onto the image plane in
different ways:
Image Plane
Center of Projection
An image(640*640)
Perspective Projection• Images are mapped onto the image plane in
different ways:
Image Plane
Center of Projection
An image(640*640)
Perspective Projection• Angle of view tells us the mapping from the
image to the image plane:
Image Plane
Center of Projection
Angle of viewField of view
Perspective Projection• Not everything will be displayed.
Image Plane
Center of Projection
Near Plane Far Plane
Frustum
Perspective Projection• The frustum of perspective projection looks
like:Center of Projection
Image Plane
Near Plane
Ortho Projection• What’s the frustum of orthographic projection?
Image Plane
Near Far
• In general, the homogeneous coordinate system is define as:
• For example,
Homogenous Coordinates
wxwywzw
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
214
0.5
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
4281
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
In 3DIn homogeneous space
• Both homogenous vectors and matrices are scalable:
Homogenous Coordinates
wxwywzw
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
wm00 wm01 wm02 wm03
wm10 wm11 wm12 wm13
wm20 wm21 wm22 wm23
wm30 wm31 wm32 wm33
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
=
m 00 m 01 m 02 m 03
m 10 m 11 m 12 m 13
m 20 m 21 m 22 m 23
m 30 m 31 m 32 m 33
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Perspective Projection• Derivation:
Image Plane
Z Axis
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
x 'y '−d1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−d
XY Plane
Perspective Projection• Derivation:
Image Plane
Z Axis
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−xd / z−yd / z
−d1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−d
XY Plane
Perspective Matrix• Derivation:
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−xd / z−yd / z
−d1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥ How?
−xd / z−yd / z
−d1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
dd
d−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
The z axis now becomes useless…
OpenGL Perspective Matrix• In practice, OpenGL uses the z axis for depth
test. Its matrix looks like this:
=
dd
a b−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−x / z−y / z
−a − b / z1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
The depth value now can be used for depth test.We will discuss this in more details later...
Vanishing Point• Given a ray:
• Its projection:
px
py
pz
1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
nx
ny
nz
1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
px + tnx
py+ tny
pz + tnz
1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
d(px + tnx)d(py+ tny)
...−(pz + tnz)
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
dd
a b−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
px + tnx
py+ tny
pz + tnz
1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
−dpx + tnx
pz + tnz
−dpy + tny
pz + tnz
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
Vanishing Point• When t goes to infinity:
• What if there is another ray:
limt→ ∞
−dpx + tnx
pz + tnz
−dpy+ tny
pz + tnz
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
−dnx
nz
−dny
nz
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
qx + tnx
qy+ tny
qz + tnz
1
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
Parallel lines meet atthe vanishing point.
Properties of Perspective Projection
• Lines are mapped to lines.• Parallel lines may not remain parallel. Instead, they
may meet at the vanishing point.• Ratios are not preserved.
• It has foreshortening effects. So it looks real.
• Distances cannot be directly measured, as in parallel projection.
Basic OpenGL Projection• Everything will be considered in the eye space:• Geometry objects have been transformed into the eye
coordinate system using the GL_MODELVIEW matrix.• You define the projection matrix in GL_PROJECTION, also in
the eye space.• OpenGL always assume that the viewing direction is the –z
direction.
• OpenGL automatically processes each vertex using GL_PROJECTION:• After projection, the frustum is converted into a canonical
view volume ( [-1, 1] in all coordinates)
OpenGL Orthographic Projection
glOrtho(left, right, bottom, top, near, far)
(left, bottom, near)
X range Y range Z range
(right, top, far)
(-1, -1, -1)
(1, 1, 1)
OpenGL Orthographic Projection
glOrtho(l, r, b, t, n, f)
• Translation so that the center is the origin.• Scaling so that the size becomes (2, 2, 2).
1 −r+l2
1 −t+ b2
1 −f+ n21
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
2r −l
2t−b
2f−n
1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
2r −l
−r+lr−l
2t−b
−t+bt−b
2f−n
−f+ nf−n1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
=
OpenGL Perspective ProjectionglFrustum(left, right, bottom, top, near, far)
Center of Projection
Near Plane(Image Plane) Far Plane
Frustum
(left, right, bottom, top)may not be centered alongthe axis
Always positive,although it’s facingthe –z direction
OpenGL Perspective ProjectiongluPerspective(fov, aspect_ratio, near, far)
Center of Projection
Near Plane(Image Plane) Far Plane
Frustumfov
ratio=image_width/image_heighttop: near*ctan(fov/2)right: near*ratio*ctan(fov/2)
Always positive,although it’s facingthe –z direction
OpenGL Perspective Projection
glFrustum(…) is less useful than gluPerspective(...).
But we can still use it for demonstration purpose next.
OpenGL Perspective Projection
−nx / z−ny / z
−n1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
nn
n−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
Projection
OpenGL Perspective Projection
−nx / z−ny / z
−a − b / z1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
nn
a b−1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥=
Projection
in (-1, 1)
in (-n, -f), why???
OpenGL Perspective Projection
−nx / z−ny / z
−a − b / z1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
−a − b / (−n) = −1 (z = n)−a − b / (− f ) = 1 (z = f )⎧⎨⎩
a =−f+ nf−n
b=−2 fnf−n
⎧
⎨⎪⎪
⎩⎪⎪
OpenGL Perspective Projection
11
−f+ nf−n
−2 fnf−n
−1
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
xyz1
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
Projection
(l/n, b/n, -1)
(r/n, t/n, 1)
OpenGL Perspective Projection
2nr −l
2nt−b
11
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
1 −r+l2n
1 −t+b2n
11
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
=
2nr−l
−r+lr−l
2nt−b
−t+bt−b
11
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
Transform
(l/n, b/n, -1)
(r/n, t/n, 1)
(-1, -1, -1)
(1, 1, 1)
TranslationScaling
OpenGL Perspective Projection
2nr −l
−r+lr−l
2nt−b
−t+bt−b
11
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥
11
−f+ nf−n
−2 fnf−n
−1
⎡
⎣
⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥
=
2nr−l
r+lr−l
2nt−b
t+ bt−b
−f+ nf−n
−2 fnf−n
−1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
Projection
(-1, -1, -1)
(1, 1, 1)
Original ProjectionTransformation
What happens after projection?• Clipping
• Viewport transformation
• Rasterization
clipping
(-1, -1)
(1, 1)viewport
(0, 0)
(800, 800)
rasterize
Depth Test• Before rasterization, all processes are done based on
vertices.
• The z coordinate at each vertex is transformed into a new z value (or called the depth value).
• During rasterization, the z value of each pixel is interpolated from vertices.
• The z value then stored in the depth buffer, for occlusion tests. (smaller z means closer).
Depth Test
• The depth buffer is part of the frame buffer:
• To enable or disable the depth buffer:
• Without the depth test, the occlusion is determined by the drawing order.
glutInitDisplayMode(GLUT_DOUBLE|GLUT_RGB|GLUT_DEPTH);
glEnable(GL_DEPTH_TEST);
glDisable(GL_DEPTH_TEST);
Common Issues
• When you set up the perspective matrix:• Near (n) cannot be zero!• f : n cannot be too large! (far cannot be too
large, or near cannot be too small.) Why?
2nr −l
r+lr−l
2nt−b
t+ bt−b
−f+ nf−n
−2 fnf−n
−1
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥