CS552: Computer Graphics Lecture 11: Orthographic Projection Recap 3D Projection View volume o Symmetric o Oblique Introduction to parallel projection Objective After completing todays lecture, students will be able to Derive mathematical expressions related to o Parallel projection o Normalize coordinate transform in case of Perspective Orthographic Oblique projection Projection Perspective Axonometric Oblique Multiview projection Parallel projection Orthogonal Projection Side Elevation view Front Elevation view Plan view View volume Parallel Projection From the third equation we can say Parallel Projection The projection formula In homogeneous representation Oblique Parallel Projection Oblique parallel projection equation Length L depends on the angle and the perpendicular distance of the point (x, y, z) from the view plane The oblique parallel projection equations Relationship with Orthogonal projection? Cavalier and Cabinet Cavalier projections Cabinet projections All lines perpendicular to the projection plane are projected with no change in length All lines perpendicular to the projection plane are projected half of its length Oblique Parallel-Projection Vector Clipping Window and Oblique Parallel- Projection View Volume Oblique Parallel-Projection Transformation Matrix What happens in case of orthographic projection? Location of the view plane? Oblique Parallel-Projection 3D Viewing pipeline 1. Translate the viewing-coordinate origin to the origin of the world coordinate system 3D Viewing pipeline 2. Apply rotations to align the view coordinate axis to world coordinate axis 3D Viewing pipeline The coordinate transformation matrix is then obtained as: Normalization transform Normalization Transformation Orthogonal Projection Normalization Transformation: Oblique Parallel Projection Normalized Perspective-Projection Transformation Mapping of the parallelepiped to a normalized view volume. Normalized Perspective-Projection Transformation The homogeneous coordinates can be obtained as: Normalized Perspective-Projection Transformation Projection coordinated are Normalization criteria InputOutput 111 Normalization parameters Normalized transformation matrix Using field-of-view angle PRP at the origin VP at the position of the near clipping plane Graphics TPA Human Face Rendering Geometric representation of Hand drawn objects Realistic rendering of indoor scenes 3D reconstruction from contours Modeling of Object Deformation Simulate cutting of soft objects Chemical formula visualizer Thank you Next Lecture: Projection Geometry