Projection in 3-D Glenn G. Chappell CHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 381 Lecture Notes Wednesday, October 29, 2003

Review: OpenGL Geometry Pipeline Model/view Transformation Lighting Projection Transformation Clipping (view-frustum) Viewport Transformation Object Coordinates Window Coordinates World Coordinates Eye Coordinates Vertex Operations Rasterization Fragment Operations Vertex enters here To framebuffer Vertices (object coord’s) Vertices (window coord’s) Fragments Fragments Convert vertex data to fragment data Depth test, etc. 29 Oct 2003 CS 381

Review: Introduction to 3-D Viewing [1/3] The model/view matrix can store a sequence of transformations. Two kinds of transformations go into model/view: modeling and viewing. The math and the code for these are identical. But conceptually they are different. Modeling transformations move objects or the scene. Viewing transformations can be thought of as moving the camera. Issues such as wide- vs. narrow-angle and perspective vs. parallel projection are not handled via model/view. With model/view we place objects & the camera in the world. Camera properties, such as those mentioned above, are set with the projection transformation. Model/view transformation Transformation #1 Transformation #2 Transformation #3 Transformation #4 29 Oct 2003 CS 381

Review: Introduction to 3-D Viewing [2/3] To move an object, put in a new transformation just before the object is drawn. To move the entire scene, put in a new transformation at the beginning of the modeling transformations. Where this is depends on how you think about things. To move the camera, put in a new transformation at the very beginning of the transformation-setting code. Moving the camera is backwards from moving objects Object forward = camera backward. Camera positioning using glTranslate*, glRotate* is often tricky. Using gluLookAt is sometimes more convenient. With gluLookAt, we specify where we look (“at point”). We can also easily use gluLookAt to set up what direction we look (“view-plane normal”). Push-pop pairs go around everything! You can never have too many of these. 29 Oct 2003 CS 381

Review: Introduction to 3-D Viewing [3/3] When we do lighting (soon!), doing things “wrong” can more easily lead to poor results. So get in the habit of doing things “right” now. Use model/view & projection properly. Draw your polygons front-side-out. If you do both lighting and scaling: glEnable(GL_NORMALIZE); Drawing an Object Conveniently Write a function that draws the object in “standard” position. In particular, the point you want to rotate about (the center of the object?) goes at the origin. To position the object, set up a transformation, then call this function. Don’t forget the push-pop pair! The same function can be used to draw multiple copies of the object. 29 Oct 2003 CS 381

Projection in 3-D: Introduction We use the projection transformation to handle camera properties. Perspective or parallel (orthogonal) projection. Wide or narrow angle. But not camera position & orientation. Using 44 matrices with the 4th-coordinate division, we can create any such projection we want. Both perspective and parallel. Including all of the classical views. Plan, elevation, isometric. 1-, 2-, and 3-point perspective. We do it all based on the synthetic camera model. Now we look at some of the mathematics. 29 Oct 2003 CS 381

Projection in 3-D: Projecting a Point [1/2] We have discussed the synthetic-camera model. Now we apply it, to determine the coordinates of a projected point. Screen Center of Projection (“eye”) (0, 0, 0) (x, y, z) –z (?, ?, ?) z = –near View Frustum z = –far 29 Oct 2003 CS 381

Projection in 3-D: Projecting a Point [2/2] Let b be the y-coordinate of the projection. Using similar triangles (outlined in red), we see that b/near = y/(–z), and so b is the value given below. We handle the x-coordinate similarly. Screen Center of Projection (“eye”) (0, 0, 0) (x, y, z) –z (x/[–z/near], y/[–z/near], –near) z = –near View Frustum z = –far 29 Oct 2003 CS 381

Projection in 3-D: A Matrix for Perspective What happens if we use the following matrix in the pipeline? Result: perspective! Is this matrix what glFrustum produces? Not quite; it deals with right, left, top, bottom, far, too. 29 Oct 2003 CS 381

Projection in 3-D: Wide vs. Narrow Angles How do we determine wide & narrow angle using glFrustum or gluPerspective? Example time … 29 Oct 2003 CS 381

More on OpenGL Matrices: Saving & Restoring Next time we will look at more advanced viewing interfaces. Zoom & pan. Flying. In order to implement these conveniently, we must be able to save and restore the value of a transformation matrix. See printmatrix.cpp. 29 Oct 2003 CS 381