Notes04.pdf
- 1. CS3162 Introduction to Computer Graphics Helena Wong, 2000 7. Three-Dimensional Viewing Viewing in 3D involves the following considerations: - We can view an object from any spatial position, eg. In front of an object, Behind the object, In the middle of a group of objects, Inside an object, etc. - 3D descriptions of objects must be projected onto the flat viewing surface of the output device. - The clipping boundaries enclose a volume of space. 7.1 Viewing Pipeline Modelling Coordinates ⇓ Modelling Transformations ⇓ World Coordinates ⇓ Viewing Transformation ⇓ Viewing Coordinates ⇓ Projection Transformation ⇓ Projection Coordinates ⇓ Workstation Transformation ⇓ Device Coordinates Explanation: Modelling Transformation and Viewing Transformation can be done by 3D transformations. The viewing-coordinate system is used in graphics packages as a reference for specifying the observer viewing position and the position of the projection plane. Projection operations convert the viewing-coordinate description to coordinate positions on the projection plane. (Involve clipping, visual-surface identification, and surface-rendering) Workstation transformation maps the coordinate positions on the projection plane to the output device.
- 2. CS3162 Introduction to Computer Graphics Helena Wong, 2000 7.2 Viewing Transformation Conversion of objection descriptions from world to viewing coordinates is equivalent to a transformation that superimposes the viewing reference frame onto the world frame using the basic geometric translate-rotate operations: 1. Translate the view reference point to the origin of the world-coordinate system. 2. Apply rotations to align the xv, yv, and zv axes (viewing coordinate system) with the world xw, yw, zw axes, respectively. 7.3 Projections There are 2 basic projection methods: 1. Parallel Projection transforms object positions to the view plane along parallel lines. A parallel projection preserves relative proportions of objects. Accurate views of the various sides of an object are obtained with a parallel projection. But not a realistic representation. 2. Perspective Projection transforms object positions to the view plane while converging to a center point of projection. Perspective projection produces realistic views but does not preserve relative proportions. Projections of distant objects are smaller than the projections of objects of the same size that are closer to the projection plane.
- 3. CS3162 Introduction to Computer Graphics Helena Wong, 2000 Parallel Projection Classification: Orthographic Parallel Projection and Oblique Projection: Orthographic parallel projections are done by projecting points along parallel lines that are perpendicular to the projection plane. Oblique projections are obtained by projecting along parallel lines that are NOT perpendicular to the projection plane. Some special Orthographic Parallel Projections involve Plan View (Top projection), Side Elevations, and Isometric Projection: The following results can be obtained from oblique projections of a cube: Interested students may find detailed discussion on the steps to do these projections in the text book. Perspective Projection The details of perspective projection are not covered in this course. Interested students can study the topic according to the text book.
- 4. CS3162 Introduction to Computer Graphics Helena Wong, 2000 7.4 View Volumes View window - A rectangular area in the view plane which controls how much of the scene is viewed. The edges of the view window are parallel to the xv and yv viewing axes. View volume - formed by the view window and the type of projection to be used. Only those objects within the view volume will appear in the generated display. So we can exclude objects that are beyond the view volume when we render the objects in the scene. A finite view volume is obtained by bounding with front plane and back plane (or the near plane and the far plane). Hence a view volume is bounded by 6 planes => rectangular parallelepiped or a frustum, for parallel projection and perspective projection respectively. Some facts: Perspective effects depend on the positioning of the center point of projection. If it is close to the view plane, perspective effects are emphasised, ie. closer objects will appear larger than more distant objects of the same size. The projected size of an object is also affected by the relative position of the object and the view plane. 'Viewing' a static view: The view plane is usually placed at the viewing-coordinate origin and the center of projection is positioned to obtain the amount of perspective desired. 'Viewing' an animation sequence: Usually the center of projection point is placed at the viewing-coordinate origin and the view plane is placed in front of the scene. The size of the view window is adjusted to obtain the amount of scene desired. We move through the scene by moving the viewing reference frame (ie. the viewing coordinate system).
- 5. CS3162 Introduction to Computer Graphics Helena Wong, 2000 7.5 Clipping The purpose of 3D clipping is to identify and save all surface segments within the view volume for display on the output device. All parts of objects that are outside the view volume are discarded. Thus the computing time is saved. 3D clipping is based on 2D clipping. To understand the basic concept we consider the following algorithm: Polygon Clipping Assuming the clip region is a rectangular area, 1. The rectangular clip region can be represented by xmin, xmax, ymin and ymax. 2. Find the bounding box for the polygon: ie. the smallest rectangle enclosing the entire polygon. 3. Compare the bounding box with the clip region (by comparing their xmin, xmax, ymin and ymax). 4. If the bounding box for the polygon is completely outside the clip region (case 2), the polygon is outside the clip region and no clipping is needed. 5. If the bounding box for the polygon is completely inside the clip region (case 1), the polygon is inside the clip region and no clipping is needed. 6. Otherwise, the bounding box for the polygon overlaps with the clip region (cases 3 and 4) and the polygon is likely to be partly inside and partly outside of the clip region. In that case, we clip the polygon against each of the 4 border lines of the clip region in sequence as follows: 1. Using the first vertex as the current vertex. If the point is in the inside of the border line, mark it as 'inside'. If it is outside, mark it as 'outside'. 2. Check the next vertex. Again mark it 'inside' or 'outside' accordingly. 3. Compare the current and the next vertices. If one is marked 'inside' and the other 'outside', the edge joining the 2 vertices crosses the border line. 4. In this case, we need to calculate where the edge intersects the border (ie. intersection between 2 lines). 5. The intersection point becomes a new vertex and we mark it as 'synthetic'. 6. Now we set the next vertex as the current vertex and the following vertex as the next vertex, and we repeat the same operations until all the edges of the polygon have been considered. 7. After the whole polygon has been clipped by a border, we throw away all the vertices marked 'outside' while keeping those marked as 'inside' or 'synthetic' to create a new polygon. 8. We repeat the clipping process with the new polygon against the next border line of the clip region. 7. This clipping operation results in a polygon which is totally inside the clip region.
- 6. CS3162 Introduction to Computer Graphics Helena Wong, 2000 7.6 Hardware Implementations Most graphics processes are now implemented in graphics chip sets. Hardware systems are now designed to transform, clip, and project objects to the output device for either 3D or 2D applications. In a typical arrangement, each of the individual chips in a chip set is responsible for geometric transformations, projection transformation, clipping, visible-surface identification, surface-shading procedure, octree representation processing, or ray-tracing etc., in a pipe-line way.
- 7. CS3162 Introduction to Computer Graphics Helena Wong, 2000 8. 3D Object Representations Methods: § Polygon and Quadric surfaces: For simple Euclidean objects § Spline surfaces and construction: For curved surfaces § Procedural methods: Eg. Fractals, Particle systems § Physically based modeling methods § Octree Encoding § Isosurface displays, Volume rendering, etc. Classification: Boundary Representations (B-reps) eg. Polygon facets and spline patches Space-partitioning representations eg. Octree Representation Objects may also associate with other properties such as mass, volume, so as to determine their response to stress and temperature etc. 8.1 Polygon Surfaces This method simplifies and speeds up the surface rendering and display of objects. For other 3D objection representations, they are often converted into polygon surfaces before rendering. Polygon Mesh - Using a set of connected polygonally bounded planar surfaces to represent an object, which may have curved surfaces or curved edges. - The wireframe display of such object can be displayed quickly to give general indication of the surface structure. - Realistic renderings can be produced by interpolating shading patterns across the polygon surfaces to eliminate or reduce the presence of polygon edge boundaries. Polygon Tables This is the specification of polygon surfaces using vertex coordinates and other attributes: 1. Geometric data table: vertices, edges, and polygon surfaces. 2. Attribute table: eg. Degree of transparency and surface reflectivity etc. Some consistency checks of the geometric data table: § Every vertex is listed as an endpoint for at least 2 edges § Every edge is part of at least one polygon § Every polygon is closed
- 8. CS3162 Introduction to Computer Graphics Helena Wong, 2000 Plane equation and visible points Consider a cube, each of the 6 planes has 2 sides: inside face and outside face. For each plane (in a right-handed coordinate system), if we look at its surface and take 3 points in counter-clockwise direction: (x1,y1), (x2,y2), and (x3,y3), we can compute 4 values: A,B,C,D as A = 3 3 2 2 1 1 z y 1 z y 1 z y 1 B = 3 3 2 2 1 1 z 1 x z 1 x z 1 x C = 1 y x 1 y x 1 y x 3 3 2 2 1 1 D = - 3 3 3 2 2 2 1 1 1 z y x z y x z y x Then, the plane equation at the form: Ax+By+Cz+D=0 has the property that: If we substitute any arbitrary point (x,y) into this equation, then, Ax + By + Cz + D < 0 implies that the point (x,y) is inside the surface, and Ax + By + Cz + D < 1 implies that the point (x,y) is outside the surface. Polygon Meshes Common types of polygon meshes are triangle strip and quadrilateral mesh. Fast hardware-implemented polygon renderers are capable of displaying up to 1,000,000 or more shaded triangles per second, including the application of surface texture and special lighting effects. 8.2 Curved Surfaces 1. Regular curved surfaces can be generated as - Quadric Surfaces, eg. Sphere, Ellipsoid, or - Superquadrics, eg. Superellipsoids These surfaces can be represented by some simple parametric equations, eg, for ellipsoid: x = rx cos s1 φ cos s2 θ, -π /2<= φ <= π/2 y = ry cos s1 φ sin s2 θ, -π <= θ <= π z = rz sin s1 φ Where s1, rx, ry, andrx are constants. By varying the values of φ and θ, points on the surface can be computed.
- 9. CS3162 Introduction to Computer Graphics Helena Wong, 2000 2. Irregular surfaces can also be generated using some special formulating approach, to form a kind of blobby objects -- The shapes showing a certain degree of fluidity. 3. Spline Representations Spline means a flexible strip used to produce a smooth curve through a designated set of points. Several small weights are distributed along the length of the strip to hold it in position on the drafting table as the curve is drawn. We can mathematically describe such a curve with a piecewise cubic polynomial function => spline curves. Then a spline surface can be described with 2 sets of orthogonal spline curves. 8.3 Sweep Representations Sweep representations mean sweeping a 2D surface in 3D space to create an object. However, the objects created by this method are usually converted into polygon meshes and/or parametric surfaces before storing. A Translational Sweep: A Rotational Sweep: Other variations: - We can specify special path for the sweep as some curve function. - We can vary the shape or size of the cross section along the sweep path. - We can also vary the orientation of the cross section relative to the sweep path.
- 10. CS3162 Introduction to Computer Graphics Helena Wong, 2000 8.4 Constructive Solid-Geometry Methods The Constructive Solid-Geometry Method (CSG) combine the volumes occupied by overlapping 3D objects using set operations: - Union - Intersection - Difference Object created by a union operation: Object created by a intersection operation: Object created by a difference operation: A CSG object can be represented with a binary tree: 8.5 Voxel Representation In voxel representation, an object is decomposed into identical cells arranged in a fixed regular grid. These cells are called voxels (volume elements), in analogy to pixels. Eg. A cylinder can be represented as follows by voxels. A '1' may represent inside the cylinder wile a '0' may represent outside of the cylinder. Alternatively we may use 8 bits to represent the transparency value. If a voxel has a value of '0', it is a fully-transparent cell. If a voxel has a value of '255', it is a non-transparent cell.
- 11. CS3162 Introduction to Computer Graphics Helena Wong, 2000 8.6 Octrees Octrees are hierarchical tree structures that describe each region of 3D space as nodes. When compared with the basic voxel representation, octrees reduce storage requirements for 3D objects. It also provides a convenient representation for storing information about object interiors. Octree encoding procedure is an extension of the quadtree encoding of 2D images: Bitmap representation of a 2D object: Quadtree encoding of a 2D object: Quadtree data structure for the above object: F = full, P = partially full, E = empty. The code of this image in quadree representation is: P PPPF FPPE PEFP FFPP FFEE FEFE FEFF FEFE FEFF FFFE. For octree representation of 3D data, the number method is as follows:
- 12. CS3162 Introduction to Computer Graphics Helena Wong, 2000 8.7 Fractals Fractal objects refer to those objects which are self-similar at all resolution. Most of the natural objects such as trees, mountains and coastlines are considered as fractal objects because no matter how far or how close one looks at them, they always appear to be somewhat similar. Fractal objects can also be generated recursively by applying the same transformation function to an object, eg. Scale down + rotate + translate. For example, a Fractal Snowflake: The name "Fractal" comes from its property: fractional dimension. Unlike Euclidean dimension, the dimensions of fractal objects may not be integers, and we call these dimensions as 'fractal dimension'. Euclidean dimensions and Fractal dimensions: - A line segment is 1D. If we divide a line into N equal parts, the parts each look like the original line scaled down by a factor of N = N1/1 . - A square is 2D. If we divide it into N parts, each part looks like the original scaled down by a factor of N1/2 . - For the fractal snowflake, when it is divided into 4 pieces, each resulting piece looks like the original scaled down by a factor of 3, so it has the dimension d such that 41/d =3. That is, d = 1.26. We may also create 3D objects in a similar way by adding a third dimension. The following example replaces each triangle of the object with a pyramid, by inserting a single point, at each step (except for the bottom face). This will result in a mountain like object, with a regular appearance. To give a more natural appearance to the created object, we usually allow some limited random variations at each level of recursion: