1 Computer Graphics Chapter 8 3D Transformations

[8]-2RM Z X Y (x,y,z) (x’,y’,z’) Initial Object Transformed Object Transformation in 3D space

[8]-3RM CartesianHomogeneous 3D Homogeneous Coordinates

[8]-4RM Translation Z X Y Offset Vector = (t x, t y, t z )

[8]-5RM Scaling About the Origin Z X Y Scale Factors = (s x, s y, s z )

[8]-6RM (x,y,z) (x’,y’,z’)  x y x y z Rotation About Z-axis

[8]-7RM (x,y,z) (x’,y’,z’)  y z x y z Rotation About X-axis

[8]-8RM (x,y,z) (x’,y’,z’)  z x x y z Rotation About Y-axis

[8]-9RM Rotation Matrices

[8]-10RM Y X Z Reflection About XY Plane

[8]-11RM Reflection about xy plane: Reflection about yz plane: Reflection about zx plane: Reflections

[8]-12RM A general invertible, linear, transformation 3D Affine Transformations

[8]-13RM Product of affine transformations is affine. Affine transformations preserve linearity of segments. Affine transformations preserve parallelism between lines. Affine transformations are invertible. Affine Transforms - Properties

[8]-14RM 3D Transformations - OpenGL

[8]-15RM 3D Viewing

[8]-16RM Planar Projections

[8]-17RM P P’ P (3D) P’ (2D) Plane of Projection Projector Center of Projection Object Point Projection Geometry

[8]-18RM Parallel and Perspective Parallel Projections:  The center of projection is at infinity.  The projectors are parallel to each other. Perspective Projections:  The center of projection is a finite point.  The projectors intersect at the center of projection.

[8]-19RM Orthographic Projection The lines of projection are parallel, and at the same time orthogonal to the plane of projection.

[8]-20RM Orthographic Projection The lines of projection are parallel, and at the same time orthogonal to the plane of projection.

[8]-21RM Orthographic Projection Orthographic Projection Matrix: Projection along z axis: Transformation:x’ = x y’ = y z-coordinate information is lost!

[8]-22RM Oblique Projection The lines of projection are parallel, but not orthogonal to the plane of projection.

[8]-23RM Oblique Projection Oblique Projection Matrix: Transformation:x’ = x+k 1 z y’ = y+k 2 z The z-coordinate value of the object point, leads to a shift of x, y coordinates of the projected point, proportional to z.

[8]-24RM Perspective Projection The projectors intersect at a Center of Projection C.

[8]-25RM Perspective Projection For convenience of deriving projection equations, we assume that the center of projection is at the origin. The object and the plane of projection are now on the negative z side. The distance D then refers to the z-coordinate of the object point (towards –z). The plane of projection has the equation z =  N.

[8]-26RM Perspective Projection Perspective Projection Matrix:

[8]-27RM Perspective Projection The z-coordinate value of the object point leads to proportional scaling along x, y directions. Projections of objects located closer to the center of projection O, appear to be larger in size compared to objects that are farther away from O.

[8]-28RM Perspective Transformation a, b are arbitrary constants. The last term z’ is referred to as the pseudo-depth. Perspective Transformation Matrix:

[8]-29RM Projections: Properties Projections map points from one space to another coordinate space of lower dimension, and hence involves loss of information. Projections are not invertible. All projection matrices are singular. All points on a projector map to the same point on the plane of projection.

[8]-30RM View Volumes A view volume refers to a region of interest in three-dimensional space which will be used to generate the display on the viewport. All points outside the view volume will not be displayed. The shape of the view volume depends on the projection used, and the limits specified by the user.

[8]-31RM View Volume - Orthographic Projn glOrtho(left, right, bottom, top, near, far);

[8]-32RM View Volume - Orthographic Projn glOrtho(left, right, bottom, top, near, far);

[8]-33RM View Volume - Perspective Projn glFrustum(left, right, bottom, top, near, far);

[8]-34RM View Volume - Perspective Projn gluPerspective(fovy, aspect, near, far);

[8]-35RM Canonical View Volumes The 3D object model is not actually clipped inside the projection view volume. The view volume is instead mapped to a canonical view volume (CVV) which is a cube that extends from  1 to +1 in each dimension, having center at the origin. The dimensions of the CVV facilitates a fast and efficient clipping.

[8]-36RM Mapping to CVV Orthographic: Perspective:

[8]-37RM