3 projection computer graphics
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This document discusses different types of projections used in computer graphics, including perspective and parallel projections. It describes orthographic projection, which projects points along the z-axis onto the z=0 plane. Perspective projection is also covered, including how it creates the effect of objects appearing smaller with distance using similar triangles. The document provides the equation for a perspective projection matrix and an example. It concludes by discussing defining a viewing region or frustum using functions like glFrustum and gluPerspective in OpenGL.
3 projection computer graphics
- 1. SBE 306: Computer Graphics Projection Dr. Ayman Eldeib Systems & Biomedical Engineering Department Spring 2019
- 2. Computer Graphics Projection Types of Projections Perspective Projection Parallel Projection “Orthographic ”
- 3. Computer Graphics Projection Simple Orthographic Projection Project all points along the z axis to the z= 0 plane 11000 0000 0010 0001 1 0 ' ' z y x y x
- 4. Computer Graphics Projection Perspective Transformations Perspective transformations do not preserve parallelism, i.e. Parallel lines not parallel; converge to single point Further objects are smaller (size, inverse distance) B A’ B’ Center of projection (camera/eye location) A
- 5. Computer Graphics Projection Perspective Projection In the real world, objects exhibit perspective foreshortening: distant objects appear smaller. The basic situation:
- 6. Computer Graphics Projection When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world How tall should this bunny be? Cont. Perspective Projection
- 7. Computer Graphics Projection The geometry of the situation is that of similar triangles. View from above: P (x, y, z)X -Z View plane d (0,0,0) x’ = ? What is x’ ? Cont. Perspective Projection
- 8. Computer Graphics Projection P (x, y, z)X -Z View plane d (0,0,0) x’ = ? z y d y z x d x ' , ' dz dz y z yd y dz x z xd x ',',' Cont. Perspective Projection
- 9. Computer Graphics Projection A Perspective Projection Matrix z y d y z x d x ' , ' dz dz y z yd y dz x z xd x ',',' d z z y x 1 1 ' ' ' d z yd z xd z y x 10 1 00 0100 0010 0001 z y x d Perspective Matrix
- 10. Computer Graphics Projection Example 0100 0100 0010 0001 d M eperspectiv dz z y x z y x d 10100 0100 0010 0001 Cont. A Perspective Projection Matrix
- 11. Computer Graphics Projection P (x, y, z)X -Z View plane d (0,0,0) x’ = ? 0100 0100 0010 0001 d M eperspectiv View plane is at –d on z coordinate (Only) last row affected (no longer 0 0 0 1) Cont. A Perspective Projection Matrix
- 12. Computer Graphics Projection Now that we can express perspective foreshortening as a matrix, we can composite it onto our other matrices with the usual matrix multiplication End result: can create a single matrix encapsulating modeling, viewing, and projection transforms Though you recall that in practice OpenGL separates the modelview from projection matrix (why?) Cont. A Perspective Projection Matrix
- 13. Computer Graphics Projection Defining a Viewing Region Most graphics API’s use two methods for defining a viewing region The viewing frustum In OpenGL glFrustum (xmin,xmax,ymin,ymax,near,far) The screen projection plane This is a special case of the viewing frustum Also called the symmetric perspective projection frustum In OpenGL gluPerspective (theta,aspect,near,far)
- 14. Computer Graphics Projection Viewing Frustum void glFrustum( GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far ) Parameters: left, right: Specify the coordinates for the left and right vertical clipping planes; bottom, top: Specify the coordinates for the bottom and top horizontal clipping planes; near, far: Specify the distances to the near and far depth clipping planes. Both distances must be positive.
- 15. Computer Graphics Projection void gluPerspective( GLdouble fovy, GLdouble aspect, GLdouble zNear, GLdouble zFar ) Parameters: Fovy: Specifies the field of view angle, in degrees, in the y direction; Aspect: Specifies the aspect ratio that determines the field of view in the x direction. The aspect ratio is the ratio of x (width) to y (height); ZNear: Specifies the distance from the viewer to the near clipping plane (always positive); ZFar: Specifies the distance from the viewer to the far clipping plane (always positive) Cont. Viewing Frustum
- 16. Computer Graphics Projection The viewing frustum defines a region of 3D space assuming that the eye is located at (0, 0, 0). A 4x4 perspective transformation maps the viewing frustum to a canonical cube Once a canonical cube is obtained 3D points are projected to 2D by dropping the z component Note perspective projection is obtained by using a perspective transformation followed by orthographic projection Cont. Viewing Frustum
- 17. Computer Graphics Projection OpenGL’s gluPerspective() command generates a slightly more complicated matrix: 2 cotwhere 0100 2 00 000 000 y farnear nearfar farnear nearfar fov f ZZ ZZ ZZ ZΖ f aspect f Cont. Viewing Frustum
- 18. Thank You