1. Shadows via Projection Glenn G. Chappell CHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 381 Lecture Notes Wednesday, November 5, 2003

2. 5 Nov 2003CS 3812 Review: Two Kinds of Interfaces Now, about that translation: In our “zoom & pan”, “driving”, the translation was done in the display function, not the saved matrix. But in “flying” the translation was in the saved matrix. Advanced interfaces can be divided into two categories: “View the world” interfaces. We look at the scene. Rotation and scaling are centered on the point we are looking at. So the translation does not go in the saved matrix. Examples Driving from above. Object manipulation. “Move in the world” interfaces. In these, we are explicitly inside the scene. Rotation is centered on the camera. So the translation goes in the saved matrix. Examples Flying. Driving, where we appear to be inside the car.

3. 5 Nov 2003CS 3813 Review: Mouse-Based Viewing [1/2] Last time we will look at two mouse-based viewing interfaces. Click-and-drag rotation. View an object and rotate it in an intuitive way with the mouse (click-and- drag). Flying with the mouse. Fly forward on mouse-button press. Turn in the direction of the mouse position. The first is a “view the world” type of interface. The second is a “move in the world type of interface. These facts have ramifications for how the interfaces are implemented. (See the previous slide.) Both of these involve rotation about a line perpendicular to a vector computed from mouse positions. First: The vector is the difference between the current and previous mouse positions. (So we needed to save the mouse position.) Second: The vector is the difference between the current mouse position and the center of the screen.

4. 5 Nov 2003CS 3814 Review: Mouse-Based Viewing [2/2] The following code was useful in both interfaces, for doing the perpendicular rotation. // perprot // Rotates about an axis in the x,y-plane, perpendicular to the given // vector (vx, vy). Rotation amount is proportional to the length of // the vector and to rotmultiplier. void perprot(double rotmultiplier, double vx, double vy) { double len = sqrt(vx*vx + vy*vy); glPushMatrix(); glRotated(rotmultiplier*len, -vy,vx,0.); glMultMatrixd( your_matrix_variable ); glGetDoublev(GL_MODELVIEW_MATRIX, your_matrix_variable ); glPopMatrix(); glutPostRedisplay(); }

5. 5 Nov 2003CS 3815 Shadows via Projection: Introduction [1/2] Now that we know something about transformation matrices, we can use them to draw shadows. Covering shadows before covering lighting is a bit odd, of course. This topic is covered in section 5.10 of the (blue) text. My presentation differs somewhat from that in the text. What is a “shadow”? Answer 1: A region where light is blocked. That’s a great answer, but to use it, we need to know something about lighting. Maybe next week … Answer 2: The projection of the shape of an object on another object. We say “shape” here, since we do not want to draw the shadow using the same colors as the original object. We know (more or less) how to project onto a plane. We can use this knowledge to draw the shadow of an object on a plane.

6. 5 Nov 2003CS 3816 Shadows via Projection: Introduction [2/2] We wish to find the matrix that projects onto a given plane from a given light position. This is essentially the synthetic-camera model. The light is at the center of projection, and the plane is the image plane. Does this mean that, with this method, objects will still cast shadows when they do not lie between the light and the plane? Yes, it does.

7. 5 Nov 2003CS 3817 Shadows via Projection: The Math [1/2] To compute the proper projection matrix, we need the light position and an equation for the plane. We represent the light position as a 4-D vector L in homogeneous form, as shown below. The equation of a plane in 3-D space can be written as Ax + By + Cz + D = 0. We represent this with another 4-D vector P, as shown below. A point V (expressed in homogeneous form) lies on the plane precisely when V·P = 0. Light Position L = (L x, L y, L z, 1) Plane Equation: Ax + By + Cz + D = 0 P = (A, B, C, D)

8. 5 Nov 2003CS 3818 Shadows via Projection: The Math [2/2] Now, let k = P·L be the dot product of P and L. The matrix that does the projecting is given below. On the left, we use matrix notation (which you may not be familiar with). What do we do with this matrix? It does projection, so we might be tempted to put it in OpenGL’s projection transformation. However, it does not set camera properties; it places objects in the world. (A shadow is an object!) So the matrix goes in model/view, after the viewing transformations, and before moving the object that casts the shadow.

9. 5 Nov 2003CS 3819 Shadows via Projection: Some Code Look at shadowproj.cpp, on the web page, for code that uses this method to draw a shadow. The light position is stored in the global lightpos. The plane is specified by putting the coordinates of points in the global planepts. Function findplane computes and returns the plane “equation” (A, B, C, D). This is called from init, where these values are stored in the global plane_eq. Function makeshadowmatrix computes and returns the shadow projection matrix. This is also called from init, where the matrix is stored in the global shadowmat. The object is drawn by drawobject. The parameter is true if the object is to be drawn in normal color, false if it is to be drawn in gray. This function is called twice: once to draw the object, and once to draw the shadow. The shadow matrix is used in function display.

10. 5 Nov 2003CS 38110 Shadows via Projection: Issues If we draw the shadow on a polygon, then the depth test might give us trouble. One solution is to disable the depth test before drawing the shadow. This only works if no part of the shadow is hidden by any previously drawn object. The shadow, being an object, is not confined to the polygon it (supposedly) is cast on. We can solve this using “stenciling”, a method for restricting drawing to certain portions of the frame buffer. Shadows are not gray; they are unlit. To do shadows properly, we draw the unshadowed portion with full lighting, and the shadowed portion using only “ambient” light. More on this when we cover lighting. Suppose we use this method to draw a shadow falling on a more complex object. For each polygon the shadow falls on, we would need to compute a separate matrix and draw the object casting the shadow. So this method is not recommended for casting shadows on complex objects. There are other shadowing methods; more on these later.