* can render 3D geometry (described using a Java 3D GeometryArray) * into a 2D Graphics context. */ public interface RenderingEngine { /** * Add a GeometryArray to the RenderingEngine. All GeometryArrays * will be rendered. */ public void addGeometry( GeometryArray geometryArray ); /** * Render a single frame into the Graphics. */ public void render( Graphics graphics, GeometryUpdater updater ); /** * Get the current Screen position used by the RenderEngine. */ public Vector3d getScreenPosition(); /** * Get the current View Angle used by the RenderEngine. View * angles are expressed in degrees. */ public Vector3d getViewAngle(); /** * Set the current View Angle used by the RenderEngine. */ public void setViewAngle( Vector3d viewAngle ); 23
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temp.normalize( ); temp.sub( light ); temp.normalize( ); cos_alpha = view.dot( temp ); intensity = (int) (lightMax * ( lightAmbient + lightDiffuse * cos_theta + lightSpecular * Math.pow( cos_alpha, lightGlossiness ))); } } } } return intensity; } 2.4 Putting it together MyJava3D The MyJava3D example defines the RenderingEngineinterface. This interface defines a simple rendering contract between a client and a 3D renderer implementation. The RenderingEngineinterface defines a simple renderer that can render 3D geometry described using a Java 3D GeometryArray. The GeometryArraycontains the 3D points and normal vectors for the 3D model to be rendered. In addition to adding GeometryArraysto the RenderingEngine(addGeometrymethod), the viewpoint of the viewer can be specified (setViewAngle), the direction of a single light can be specified (setLightAngle), the scaling factor to be applied to the model can be varied (setScale), and the size of the rendering screen defined (setScreenSize). To render all the GeometryArraysadded to the RenderingEngineusing the current light, screen, scale, and view parameters, clients can call the render method, supplying a Graphicsobject to render into, along with an optional GeometryUpdater. The GeometryUpdaterallows a client to modify the positions of points or rendering parameters prior to rendering. From AwtRenderingEngine.java /** * Definition of the RenderingEngine interface. A RenderingEngine 22
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Figure 2.6 MyJava3D rendering without light intensity calculations The computeIntensitymethod calculates the color intensity to use when rendering a triangle. It accepts a GeometryArraycontaining the 3D points for the geometry, an index that is the first point to be rendered, and a count of the number of points (vertices) that compose the item to be rendered. The method then computes the average normal vector for the points to be rendered by inspecting the normal vectors stored within the GeometryArray. For a triangle (three vertices) this will be the vector normal to the plane of the surface. The angle between the surface normal and the viewer is then calculated (beta). If the cosine of this angle is less than or equal to zero, the facet cannot be seen by the viewer and an intensity of zero will be returned. Otherwise, the method computes the angle between the light source position vector and the surface normal vector of the surface (theta). If the cosine of this angle is less than or equal to zero, none of the light from the light source illuminates the surface, so its light intensity is set to that of the ambient light. Otherwise, the surface normal vector is multiplied by the cosine of theta, the resulting vector is normalized, and then the light vector subtracted from it and the resulting vector normalized again. The angle between this vector and the viewer vector (alpha) is then determined. The intensity of the surface is the sum of the ambient light, the diffuse lighting from the surface multiplied by the cosine of the theta, and the specular light from the surface multiplied by the cosine of alpha raised to the glossiness power. The last term is the Phong shading, which creates the highlights that are seen in illuminated curved objects. Note that in this simple MyJava3D example only one light is being used to illuminate the scene in Java3D, OpenGL, or Direct3D many lights can be positioned within the scene and the rendering engine will compute the combined effects of all the lights on every surface. Please refer to chapter 10 for a further discussion of lighting equations and example illustrations created using Java 3D. From AwtRenderingEngine.java private int computeIntensity( GeometryArray geometryArray, int index, int numPoints ) { int intensity = 0; if ( computeIntensity != false ) { 20
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// if we have a normal vector, compute the intensity // under the lighting if ( (geometryArray.getVertexFormat( ) GeometryArray.NORMALS) == GeometryArray.NORMALS ) { double cos_theta; double cos_alpha; double cos_beta; for( int n = 0; n 0.0 ) { cos_theta = surf_norm.dot( light ); if ( cos_theta <= 0.0 ) { intensity = (int) (lightMax * lightAmbient); } else { temp.set( surf_norm ); temp.scale( (float) cos_theta ); 21
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An uncomplicated depth sort is easy to implement; however, once you start using it you will begin to see strange rendering artifacts. The essential problem comes down to how you measure the distance a triangle is from the viewer. Perhaps you would Take the average distance of each of the three vertices Take the distance of the centroid of the triangle With either of these simple techniques, you can generate scenes with configurations of triangles that render incorrectly. Typically, problems occur when: Triangles intersect Centroid or average depth of the triangle is not representative of the depth of the corners Complex shapes intersect Shapes require splitting to render correctly For example, figure 2.5 shows some complex configurations of triangles that cannot be depth sorted using a simple algorithm. Figure 2.5 Interesting configurations of triangles that are challenging for depth-sorting algorithms The depth of an object in the scene can be calculated if the position of the object is known and the position of the viewer or image plane is known. It would be computationally intensive to have to re-sort all the triangles in the scene every time an object or the viewer s position changed. Fortunately, binary space partition (BSP) trees can be used to store the relative positions of the object in the scene such that they do not need to be re-sorted when the viewpoint changes. BSP trees can also help with some of the complex sorting configurations shown earlier. Depth buffer (Z-buffer) In contrast to the other two algorithms, the Z-buffer technique operates in image space. This is conceptually the simplest technique and is most commonly implemented within the hardware of 3D graphics cards. If you were rendering at 640 480 resolution, you would also allocate a multidimensional array of integers of size 640 480. The array (called the depth buffer or Z-buffer) stores the depth of the closest pixel rendered into the image. As you render each triangle in your scene, you will be drawing pixels into the frame-buffer. Each pixel has a color, and an xy-coordinate in image space. You would also calculate the z-coordinate for the pixel and update the Z-buffer. The values in the Z-buffer are the distance of each pixel in the frame from the viewer. 18
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Before actually rendering a pixel into the frame-buffer for the screen display, inspect the Z-buffer and notice whether a pixel had already been rendered at the location that was closer to the viewer than the current pixel. If the value in the Z-buffer is less than the current pixel s distance from the viewer, the pixel should be obscured by the closer pixel and you can skip drawing it into the frame-buffer. It should be clear that this algorithm is fairly easy to implement, as long as you are rendering at pixel level; and if you can calculate the distance of a pixel from the viewer, things are pretty straightforward. This algorithm also has other desirable qualities: it can cope with complex intersecting shapes and it doesn t need to split triangles. The depth testing is performed at the pixel level, and is essentially a filter that prevents some pixel rendering operations from taking place, as they have already been obscured. The computational complexity of the algorithm is also far more manageable and it scales much better with large numbers of objects in the scene. To its detriment, the algorithm is very memory hungry: when rendering at 1024 800 and using 32-bit values for each Z-buffer entry, the amount of memory required is 6.25 MB. The memory requirement is becoming less problematic, however, with newer video cards (such as the nVidia Geforce II/III) shipping with 64 MB of memory. The Z-buffer is susceptible to problems associated with loss of precision. This is a fairly complex topic, but essentially there is a finite precision to the Z-buffer. Many video cards also use 16-bit Z-buffer entries to conserve memory on the video card, further exacerbating the problem. A 16-bit value can represent 65,536 values so essentially there are 65,536 depth buckets into which each pixel may be placed. Now imagine a scene where the closest object is 2 meters away and the furthest object is 100,000 meters away. Suddenly only having 65,536 depth values does not seem so attractive. Some pixels that are really at different distances are going to be placed into the same bucket. The precision of the Z-buffer then starts to become a problem and entries that should have been obscured could become randomly rendered. Thirty-two-bit Z-buffer entries will obviously help matters (4,294,967,296 entries), but greater precision merely shifts the problem out a little further. In addition, precision within the Z-buffer is not uniform as described here; there is greater precision toward the front of the scene and less precision toward the rear. When rendering using a Z-buffer, the rendering system typically requires that you specify a near and a far clipping plane. If the near clipping plane is located at z = 2 and the far plane is located at z = 10, then only objects that are between 2 and 10 meters from the viewer will get rendered. A 16-bit Z-buffer would then be quantized into 65,536 values placed between 2 and 10 meters. This would give you very high precision and would be fine for most applications. If the far plane were moved out to z = 50,000 meters then you will start to run into precision problems, particularly at the back of the visible region. In general, the ratio between the far and near clipping (far/near) planes should be kept to below 1,000 to avoid loss of precision. You can read a detailed description of the precision issues with the OpenGL depth buffer at the OpenGL FAQ and Troubleshooting Guide (http://www.frii.com/~martz/oglfaq/depthbuffer.htm). 2.3 Lighting effects MyJava3D includes some simple lighting calculations. The lighting equation sets the color of a line to be proportional to the angle between the surface and the light in the scene. The closer a surface is to being perpendicular to the vector representing a light ray, the brighter the surface should appear. Surfaces that are perpendicular to light rays will absorb light and appear brighter. MyJava3D includes a single white light and uses the Phong lighting equation to calculate the intensity for each triangle in the model (figure 2.6). 19
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Figure 2.4 Compare the output from Java 3D (left) with the output from MyJava3D (right) 2.2.3 Drawing filled triangles Java 3D rendered the hand as an apparently solid object. We cannot see the triangles that compose the hand, and triangles closer to the viewer obscure the triangles further away. You could implement similar functionality within MyJava3D in several ways: Hidden surface removal You could calculate which triangles are not visible and exclude them from rendering. This is typically performed by enforcing a winding order on the vertices that compose a triangle. Usually vertices are connected in a clockwise order. This allows the graphics engine to calculate a vector that is normal (perpendicular) to the face of the triangle. The triangle will not be displayed if its normal vector is pointing away from the viewer. This technique operates in object space as it involves mathematical operations on the objects, faces, and 2 edges of the 3D objects in the scene. It typically has a computational complexity of order n where n is the number of faces. This quickly becomes complicated however as some triangles may be partially visible. For partially visible triangles, an input triangle has to be broken down into several new wholly visible triangles. There are many good online graphics courses that explain various hidden-surface removal algorithms in detail. Use your favorite search engine and search on hidden surface removal and you will find lots of useful references. Depth sorting (Painter s algorithm) The so-called Painter s algorithm also operates in object space; however, it takes a slightly different approach. The University of North Carolina at Chapel Hill Computer Science Department online course Introduction to Computer Graphics (http://www.cs.unc.edu/~davemc/Class/136/) explains the Painter s algorithm (http://www.cs.unc.edu/~davemc/Class/136/Lecture19/Painter.html). The basic approach for the Painter s algorithm is to sort the triangles in the scene by their distance from the viewer. The triangles are then rendered in order: triangle furthest away rendered first, closest triangle rendered last. This ensures that the closer triangles will overlap and obscure triangles that are further away. 17
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Figure 2.3 The positions of some projected points Plotting these points by hand using a 2D graphics program (figure 2.3), you can see that they seem to make sense. Projecting the point 0,0,0 places a point at the center of the screen (160,120). While you have symmetry about the corners of the cube, increasing the Z-coordinate appears to move the two opposing corners (1,1,1 and -1,-1,1) closer to the viewer. Taking a look at the projectPointfunction again, you can see it uses the following parameters: Input point x, y, and z coordinates Center of the screen Sin and cosine of the viewer s angle of view Distance of the screen from the viewer Model scaling factor This very simple projection function is adequate for simple 3D projection. As you become more familiar with Java 3D, you will see that it includes far more powerful projection abilities. These allow you to render to stereo displays (such as head-mounted displays) or perform parallel projections. (In parallel projections, parallel lines remain parallel after projection.) 2.2.2 Comparing output Look at the outputs from MyJava3D and Java 3D again (figure 2.4). They are very different so Java 3D must be doing a lot more than projecting points and drawing lines: Triangles are drawn filled; you cannot see the edges of the triangles. Nice lighting effect can be seen in the curve of the hand. Background colors are different. Performance is much better measured by comparing the number of frames rendered per second. 16
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From AwtRenderingEngine.java private int xScreenCenter = 320/2; private int yScreenCenter = 240/2; private Vector3d screenPosition = new Vector3d( 0, 0, 20 ); private Vector3d viewAngle = new Vector3d( 0, 90, 180 ); private static final double DEG_TO_RAD = 0.017453292; private double modelScale = 10; CT = Math.cos( DEG_TO_RAD * viewAngle.x ); ST = Math.sin( DEG_TO_RAD * viewAngle.x ); CP = Math.cos( DEG_TO_RAD * viewAngle.y ); SP = Math.sin( DEG_TO_RAD * viewAngle.y ); public void projectPoint( Point3d input, Point3d output ) { double x = screenPosition.x + input.x * CT - input.y * ST; double y = screenPosition.y + input.x * ST * SP + input.y * CT * SP + input.z * CP; double temp = viewAngle.z / (screenPosition.z + input.x * ST * CP + input.y * CT * CP - input.z * SP ); output.x = xScreenCenter + modelScale * temp * x; output.y = yScreenCenter - modelScale * temp * y; output.z = 0; } Let s quickly project some points using this routine to see if it makes sense. The result of running seven 3D points through the projectPoint method is listed in table 2.1. CT: 1 ST: 0 SP: 1 CP: 0 Table 2.1 Sample output from the projectPoint method to project points from 3D-world coordinates to 2D-screen coordinates WX WY WZ SX SY 1 1 0 250 30 -1 1 0 70 30 1 -1 0 250 210 -1 -1 0 70 210 0 0 0 160 120 1 1 1 255 25 -1 -1 1 65 215 15
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Consider the simplest series of operations that must take place to convert the 3D model data into a rendered image: 1. Load the 3D points that compose the vertices (corners) of each triangle. The vertices are indexed so they can be referenced by index later. 2. Load the connectivity information for the triangles. For example, a triangle might connect vertices 2, 5, and 7. The actual vertex information will be referenced using the information and indices established in step 1. 3. Perform some sort of mathematical conversion between the 3D coordinates for each vertex and the 2D coordinates used for the pixels on the screen. This conversion should take into account the position of the viewer of the scene as well as perspective. 4. Draw each triangle in turn using a 2D graphics context, but instead of using the 3D coordinates loaded in step 1, use the 2D coordinates that were calculated in step 3. 5. Display the image. That s it. Steps 1, 2, 4, and 5 should be straightforward. Steps 1 and 2 involve some relatively simple file I/O, while steps 4 and 5 use Java s AWT 2D graphics functions to draw a simple line into the screen. Step 3 is where much of the work takes place that qualifies this as a 3D application. In fact, in the MyJava3D example application, we cheat and use some of the Java 3D data structures. This allows us to use the existing Lightwave OBJ loader provided with Java 3D to avoid doing the tiresome file I/O ourselves. It also provides useful data structures for describing 3D points, objects to be rendered, and so on. 2.2 Projecting from 3D world coordinates to 2D screen coordinates Performing a simple projection from 3D coordinates to 2D coordinates is relatively uncomplicated, though it does involve some matrix algebra that I shan t explain in detail. (There are plenty of graphics textbooks that will step you through them in far greater detail than I could here.) There are also many introductory 3D graphics courses that cover this material online. A list of good links to frequently asked questions (FAQs) and other information is available from 3D Ark at http://www.3dark.com/resources/faqs.html. If you would like to pick up a free online book that discusses matrix and vector algebra related to 3D graphics, try Sbastien Loisel s Zed3D, A compact reference for 3D computer graphics programming. It is available as a ZIP archive from http://www.math.mcgill.ca/~loisel/. If you have some money to spend, I would recommend picking up the bible for these sorts of topics: Computer Graphics Principles and Practice, by James Foley, Andries van Dam, Steven Feiner, and John Hughes (Addison-Wesley, 1990). 2.2.1 A simple 3D projection routine Here is my simple 3D-projection routine. The projectPointmethod takes two Point3dinstances, the first is the input 3D-coordinate while the second will be used to store the result of the projection from 3D to 2D coordinates (the z attribute will be 0). Point3dis one of the classes defined by Java 3D. Refer to the Java 3D JavaDoc for details. Essentially, it has three public members, x, y, and z that store the coordinates in the three axes. 14
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