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<HMTL><HEAD><TITLE> Tips for Assignment Three</TITLE><!-- describe the document, avoid context sensative descriptions ><meta name="description" value="Tips for Assignment Three"<!-- keywords for the document ><meta name="keywords" value="byu, graphics, computer, syllabus"><!-- should be "document" unless providing a search, then "service" ><meta name="resource-type" value="document"><!-- use global for documents to be indexed outside BYU ><meta name="distribution" value="local"></HEAD><BODY><H1>Tips for Assignment Three</H1>This page is a collection of hints about how to approach Assignment Three. It is designed to make your life easier by explaining an object oriented approach to the assignment. It is not intended to stifle any of your own creativity in designing classes or algorithms. It is also not a comprehensive list of all algorithms you might use.<BR><BR><STRONG>Note:</STRONG>There is a change to the assignment sheet giving you up to 30 points available on the Rendering section. <HR><H2>Classes</H2>Having a few data structures with abstracted operations can make life easier. Though a deep object oriented style is used here, it need not be strictly followed.Overloaded operators are not absolutely necessary; member functions can do as well.It is also not absolutely necessary that the abstract data types be classes; they can be structures if you are unfamiliar with C++. Nonetheless, I believe that these things are easier to code and understand in C++, and you will probablyhave more fun. <HR><H3>Vector:</H3><UL><LI><STRONG>Data:</STRONG><UL><LI> Array of four floats (or doubles). You can make life easier by making thempublic.</UL><LI><STRONG>Operations:</STRONG> <UL><LI><STRONG>Constructor:</STRONG> Initialize fourth coordinate to one (or zero). <LI><STRONG>Length:</STRONG> Returns the magnitude of the vector.<LI><STRONG>Unit:</STRONG> Normalizes the vector to unit length. ( or returns a vector which is normalized to unit length.)<LI><STRONG>v:</STRONG> Quick Way to send the vector to the graphics system. Notethat although the array has four elements, there is absolutely no harm in calling v2f() on it.</UL><LI><STRONG>Operators:</STRONG><UL><LI><STRONG>dot:</STRONG> dot product of two vectors; a scalar value.<LI><STRONG>cross:</STRONG> cross product of two vectors; a third vector. Explanation of a cross product is in your appendix. <LI><STRONG>+ and -:</STRONG> add or subtract two vectors; negate a vector.<LI><STRONG>* and /:</STRONG> multiply a matrix times a vector; mutliply or divide by a scalar.</UL></UL> Take a look at the object file format information available <!WA0><A HREF="http://www.cs.byu.edu/courses/cs455/images.html">here</A>.The vector is an important class for storing three dimensional coordinates. The lastcoordinate in the array is a homogeneous coordinate. This class is adequate for storing both the 3D coordinates for a vertex and the normal at that vertex. Theseare very small but important data structures.<BR><PRE> Point [ X ] Direction [ dX ] Vector [ Y ] Vector [ dY ] [ Z ] (normal) [ dZ ] [ 1 ] [ 0 ]</PRE>The homogeneous coordinate should be 1 if the vector represents a point in threedimensional space and 0 if it represents a normal vector, which has direction but no position. This clever little trick will prevent the direction vectors from being mangled by transformations; they can be rotated and scaled but not translated.They will point in the right direction if they are transformed by the same matrix that transforms the points and then normalized to unit length. NEVER divide by thehomogeneous coordinate of a direction vector. You can make point and direction vectors separate classes or just remember to treat them a little differently. <HR><H3>Matrix:</H3><UL><LI><STRONG>Data:</STRONG><UL><LI>Four by Four Array of floats( or doubles). </UL><LI><STRONG>Operations:</STRONG> <UL><LI> <STRONG>Constructor:</STRONG> calls Identity.<LI> <STRONG>Identity:</STRONG> reset to Identity matrix.<PRE> [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 0 1 ] </PRE><LI> <STRONG>Translate(vector v):</STRONG> Premultiply itself by a matrix that will translate by vector v.<LI> <STRONG>Scale( Sx, Sy, Sz):</STRONG> Premultiply by a matrix that will Scale in X by Sx, in Y by Sy, and in Z by Sz. <LI> <STRONG>Rotate( Axis, angle):</STRONG> Premultiply by a matrix that will rotatearound the specified axis. Note that this can also be done with the next rotate function. <LI> <STRONG>Rotate(vector u, vector v, vector n):</STRONG> Premultiply by a matrix that will rotateinto a new set of axis, as explained on page 438.<LI> <STRONG>Perspective(dist d):</STRONG> Premultiply by a matrix that will perspectivizeviewing at a distance of d from the view plane. Despite all appearance, this is not ahard matrix to build. It's simplicity comes from setting zvp=0 on page 445. I've alsomade a modification so that the z-value will not be lost(keep it; you'll need it). Otherwise,the third position on the diagonal would be zero. <PRE> [ 1 0 0 0 ] [ 0 1 0 0 ] [ 0 0 1 0 ] [ 0 0 -1/d 1 ] </PRE></UL><LI> <STRONG>Operators:</STRONG><UL><LI><STRONG>*:</STRONG> multiply a matrix times a vector; Concatenate two matrices.</UL></UL> <STRONG>NOTE:</STRONG> It will not be acceptable to take shortcuts in concatenatingmatrices or multiplying them times vectors. It is likewise no longer acceptable to ignore the homogeneous coordinate. Any multiplication should be done in full.<HR><H3>Others:</H3>There are other classes you may consider having:<UL><LI><STRONG>Vertex:</STRONG> Contains both point vector and associated normal direction vector.<LI><STRONG>Polygon:</STRONG> Contains a list or array of Vertex. <LI><STRONG>PolygonGroup:</STRONG> Contains a bunch of polygons, linked by a common color. <LI><STRONG>Zbuffer:</STRONG> Does Depth Buffer testing, which you will learn in chapter 15.<LI><STRONG>LightSource:</STRONG> You'll need at least one point light source to do the shading in Chapter 14. <LI><STRONG>Rgbhue:</STRONG> You might find it useful in shading to have RGB triplesor colors with even more properties for color calculations.</UL><HR>With the matrix and vector class, you can get through quite a bit of theassignment in a less painful fashion. You will see how these operations areuseful as we proceed with the assignment.<HR><H2>Steps</H2>In this assignment, I found that it was helpful to think of the steps I wouldneed to go through to do the assignment. This is starting from the point where youhave data read in from a file and you are ready to send things through the rendering pipeline.<HR><H3>Part One: Creating a Viewing Matrix </H3>You will need to create a matrix which will do all the transformations necessary.This algorithm is designed for perspective projection from an arbitrary angle.To do this, you should only need a few parameters, which you can read from a fileor set somehow. In any case, they must be changeable. These parameters will be a view reference point (point vector), view point (point vector), and a viewup vector (direction vector). The view up vector is the V vector mentioned in 12-2and is the orientation of the head.<BR><BR>There are certain steps to take in creating this matrix:<OL><LI>Translate Vp to the origin. Trans(-vp).<LI>Rotate the world so that the viewing is down the positive Z-axis. You cando this by rotating about three axes, or make life simpler by using the equations on page 438. The N on page 438 is the vector Vrp-Vp, while V is theview up vector that you have specified. You now have a parallel view.<LI>Perspectivize the view, using the length of Vrp-Vp as d. The view plane willnow look as so.<PRE> ___________ | +d| | | | | -d_____|_____+d | 0| | | | | |___-d|_____| </PRE>Assuming a 45 degree field of view angle, this is what it should looklike right now. If you desire a narrower field of view (30 deg is nice), tryscaling right here by 1/tan(pheta) in x and y.<BR>Because the screen coordinates are positive pixel values, if we were to renderthis plane now, we would see only:<PRE> ______ +d| | | | |_____+d </PRE>We need to fix this by going through steps 4 and 5.<LI>Scale the drawing to the viewpoint, taking into account the angle of the field of view. Scale the z value by the same amount, just to make it more numerically stable. <LI>Translate so that the origin of the view plane is centered in the viewport.These last two transformations should change the figure above intothe one below: <PRE> (XMAX,YMAX) ___________ | | | | | | | | | | |___________| (0,0)</PRE>which looks amazingly like the window on your screen.<BR>If you are using a windowing system </OL>A Graphical Representation of this chain is below:<PRE>[ Trans ][ Scale ][ Persp ][ Rot ][ Trans ][Center ][ to ][ d= ][ Align ][ Vp ][in View][ View ][ |Vrp-Vp|][ View on ][ to ][ Port ][ Port ][ ][ Z axis ][ Origin ]</PRE>Unfortunately, the normal direction vector won't be immune to the last translation performed, as I originally thought. Therefore, theeasiest way to handle the normal vectors is to create another matrix thatlooks like this:<PRE> [ Rot ] [ Align ] [ View on ] [ Z axis ]</PRE>Note that this has just one of the matrices from above. If you do a sheer ornonuniform scaling it will need to be in this matrix as well. <HR><H3>Part Two: Rendering</H3>There are certain amount of steps you go through to rendera model on screen.<UL><LI>Do a few steps for initialization. <UL><LI>Transform your light source(s) using the same transform you are about touse on your polygons. <LI>Initialize your z-buffer to a very large negative value.</UL><LI>For each polygon in your model:<UL><LI>Transform the vertices and their associated normal direction vectors by thematrices that you have laboriously computed in part one. Remember to normalize the normal vectors to unit length and to divide the point vectors by their homogeneouscoordinate. <LI>Check to see if the polygon points away from you(dZ<0) if doing back-face detection. (The normal to the polygon can be calculated by (v2-v1)X(vn-v1) where v1 .. vn are the vertices of the polygon).<LI>Place their transformed edges into the polygon scanfill's edge table.<LI>While Rendering a polygon:<UL><LI>Interpolate Z values and normal vectors across x and y.<LI>For each pixel:<OL><LI>Test against z-buffer<LI>If Step 1 passes, shade pixel according to equation 14-9 on page 504. You willneed the point vector(including the z value), and the normal direction vector tomake this calculation.</OL></UL></UL></UL><HR><H3>Change to Polygon Scan Conversion</H3> Accomodating three dimensions will require you to make some changes to your Polygon Scan conversion routines. As near as I have been able to determine, thefollowing is a summary of those changes:<UL><LI> It must be compatible with your vector class.<LI> In additon to a dxPerScan, it should have a dzPerScan and a dNormalDirectionVectorPerScan, which should also be calculated in makeEdgeRec andadded to the z value and Normal Vector in updateActiveList. <LI>The fill scan function will no longer be as simple as shown. It will be helpfulin this function to have a dzPerxpixel and a dNormalDirectionVectorPerxpixel which will be added to the z value and Normal Vector each time x is incremented. In each step in x, the point vector will be checked against the Zbuffer and then shaded if it passes. </UL><HR><H3>Shading</H3>There should be a few things to keep in mind while shading. The equation which youwill probably use for shading is equation 14-15 on page 508. You will do this calculation for the R, G, and B components.<BR>Note that there seem to be a lot of variables in this equation:<UL> <LI>ka,kd,and ks are the ambient, diffuse, and specular coefficients. They areproperties of the material and should be fractions less than one. You can think of them as the percentage of each kind of light taken into account. ka=.3, kd=.4,and ks=.3 signifies that the surface is lighted by 30% ambient light, 40% diffuse light, and 30% specular light. You will need to come up with these. <LI>Sdb and Ssb are the diffuse and specular color of the material. The color that you read in from a file, if any, is the diffuse color. You can come up with an appropriate specular color. For a plastic type object, this is the same as the diffuse color. For a metallic object, it would be closer to white.<LI>Iab is the color of the ambient light, while Ilbi is the color of each light source. These are global parameters. <LI>fi(d) is the attenuation of each light source. One light source can beadequate for purposes of this assignment. Although there are several complicatedequations for light attenuation on page 506, a simple one that works well isf(d)=a1/d. You may need to experiment with a1. Mine was WINSIZE/2.<LI>ns is an integer exponent (called the phong constant). This is also a property of the material that you should come up with. Page 503 shows the effect of thisconstant. Larger numbers result in sharper highlights.<LI> There are some vectors mentioned in the equation( N, H, L). N is the unitizednormal direction vector that you have laboriously preserved for this calculation, while L is a unitized vector that points out to the light source. With N and L, you can easily calculate any other vector, R or H, that you might need by using the equations on page 503. </UL><HR><H3>Z Buffer</H3>A Z buffer is not hard to do. There is just one note here about the Zbuffer, whichis a big array of z values. Each z value should be initialized to a large negativevalue, not 0 as shown on page 473. It will then work properly to compare the z valueof a pixel. If the z value is greater than the existing z value, replace the z valueand draw the pixel. This is a simple test because it is pixel by pixel. <HR>Questions about the material on this page? Please <!WA1><A HREF="mailto:mellor@cs.byu.edu">ask.</A><HR>Unpublished Copyright Nathan Mellor, 1996. Reproduction permitted for personaleducational use. </BODY></HTML>⌨️ 快捷键说明
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