3-D Transformation in Computer Graphics
- 1. 4. UNIT-V THREE DIMENSIONAL TRANSFORMATIONS BASIC TRANSFORMATION Geometric transformations and object modeling in three dimensions are extended from two- dimensional methods by including considerations for the z-coordinate. Translation This produces a translation in the opposite direction and the product of a translation matrix and its inverse produces the identity matrix. Rotation To generate a rotation transformation for an object an axis of rotation must be designed to rotate the object and the amount of angular rotation is also be specified. In a three dimensional homogeneous coordinate representation, a point or an object is translated from position P = (x, y, z) to position () P' = x',y',z with the matrix operation. Parameters tx, ty and tz specifying translation distances for the coordinate directions x, y and z are assigned any real values. The matrix representation in equation (1) in equivalent to the three equations. Inverse of the translation matrix in equation (1) can be obtained by negating the translation distances tx, ty and tz.
- 2. Positive rotation angles produce counter clockwise rotations about a coordinate axis. Co-ordinate Axes Rotations The 2D z axis rotation equations are easily extended to 3D. Substituting permutations (5) in Equation (2), we get the equations for an x-axis rotation: Parameters θ specifies the rotation angle. In homogeneous coordinate form, the 3D z axis rotation equations are expressed as Transformation equations for rotation about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters x, y and z in equation (2) i.e., we use the replacements.
- 3. which can be written in the homogeneous coordinate form Rotation of an object around the x-axis is demonstrated in Figure 4.8. Cyclically permuting coordinates in equation (6) give the transformation equation for a y axis rotation. The matrix representation for y-axis rotation is An example of y axis rotation is shown in Figure 4.9.
- 4. An inverse rotation matrix is formed by replacing the rotation angle θ by –θ. Negative values for rotation angles generate rotations in a clockwise direction, so the identity matrix is produced when any rotation matrix is multiplied by its inverse. Since only the sine function is affected by the change in sign of the rotation angle, the inverse matrix can also be obtained by interchanging rows and columns. (i.e.) we can calculate the inverse of any rotation matrix R by evaluating its transpose (R–1 = RT). General Three Dimensional Rotations A rotation matrix for any axis that does not coincide with a coordinate axis can be set up as a composite transformation involving combinations of translations and the coordinate axes rotations. We obtain the required composite matrix by (1) Setting up the transformation sequence that moves the selected rotation axis onto one of the coordinate axes. (2) Then set up the rotation matrix about that coordinate axis for the specified rotation angle. (3) Obtaining the inverse transformation sequence that returns the rotation axis to its original position. In the special case where an object is to be rotated about an axis that is parallel to one of the coordinate axes, we can attain the desired rotation with the following transformation sequence (1) Translate the object so that the rotation axis coincides with the parallel coordinate axis. (2) Perform the specified rotation about that axis. (3) Translate the object so that the rotation axis is moved back to its original position. When an object is to be rotated about an axis that is not parallel to one of the coordinate axes, we need to perform some additional transformations. In such case, we need rotations to align the axis with a selected coordinate axis and to bring the axis back to its original orientation.
- 5. Given the specifications for the rotation axis and the rotation angle, we can accomplish the required rotation in five steps: (1) Translate the object so that the rotation axis passes through the coordinate origin. (2) Rotate the object so that the axis of rotation coincides with one of the coordinate axes. (3) Perform the specified rotation about that coordinate axis. (4) Apply inverse rotations to bring the rotation axis back to its original orientation. (5) Apply the inverse translation to bring the rotation axis back to its original position. Scaling The matrix expression for the scaling transformation of a position P = (x, y, z) relative to the coordinate origin can be written as Where scaling parameters sx, sy and sz are assigned any position values. Explicit expressions for the coordinate transformations for scaling relative to the origin are Scaling an object changes the size of the object and repositions the object relative to the coordinate origin. If the transformation parameters are not equal, relative dimensions in the object are changed. The origin shape of the object is preserved with a uniform scaling (sx = sy = sz). (Figure 4.10) shows the result of scaling an object uniformly with each scaling parameter set to 2.
- 6. Scaling with respect to a selected fixed position (xf, yf, zf) can be represented with the following transformation sequence: (1) Translate the fixed point to the origin. (2) Scale the object relative to the coordinate origin using equation (11). (3) Translate the fixed point back to its original position. This sequence of transformation is shown in Figure 4.11. The matrix representation for an arbitrary fixed point scaling can be expressed as the concatenation of the translate-scale-translate transformations are
- 7. Inverse scaling matrix m formed by replacing the scaling parameters sx, sy and sz with their reciprocals. The inverse matrix generates an opposite scaling transformation, so the concatenation of any scaling matrix and its inverse produces the identity matrix. OTHER TRANSFORMATIONS Reflections Reflections about other planes can be obtained as a combination of rotations and coordinate plane reflections. Shears Shearing transformations are used to modify object shapes. They are also used in three dimensional viewing for obtaining general projections transformations. The following transformation produces a z-axis shear. Parameters a and b can be assigned any real values. This transformation matrix is used to alter x and y coordinate values by an amount that is proportional to the z value, and the z coordinate will be changed. Boundaries of planes that are perpendicular to the z axis are shifted by an amount proportional to A 3D reflection can be performed relative to a selected reflection axis or with respect to a selected reflection plane. Reflection relative to a given axis are equivalent to 180° rotations about the axis. When the reflection plane in a coordinate plane (either xy, xz or yz) then the transformation can a conversion between left-handed and right-handed systems. An example of a reflection that converts coordinate specifications from a right handed system to a left-handed system is shown in Figure 4.12. This transformation changes the sign of z coordinates, leaves the x and y coordinate values unchanged. The matrix representation for this reflection of points relative to the xy plane is
- 8. z figure 4.13 shows the effect of shearing matrix on a unit cube for the values a = b = 1. CLASSIFICATION OF VISIBLE SURFACE DETECTION ALGORTHIM Visible face detection algorithms are broadly classified according to whether they deal with object definitions directly or with their projected images. These two approaches are called object-space methods and image space methods. An object-space method compares objects and parts of objects to each other within the scene definition to determine which surfaces, as a whole, we should label as visible. In a image-space algorithm, visibility is decided point by point at each pixel position on the projection plane. Most visible- surface algorithms use image-space methods, although object-space methods can be used effectively locate visible surfaces in some cases. Line- display algorithms, on the other hand, generally use object-space methods to identify visible lines in wire-frame displays, but many image-space visible-surface algorithms can be adapted easily to visible-line detection. BACK-FACE DETECTION A fast and simple object-space method for identifying the back faces of a polyhedron is based on the „inside-outside‟ tests. A point (x, y, z) is “inside” a polygon surface with plane parameters A, B, C and D if Ax + By+ Cz + D < 0 When an inside point is along the line of sight to the surface, the polygon must be a backface. This test is done by considering the normal N to a polygon surface, which has Casterian components (A, B, C).
- 9. If V is a vector in the viewing direction from the eye (or “camera”) position, then the polygon is a backface if V.N > 0 If object descriptions have been connected to projection coordinates and the viewing direction is parallel to the viewing Zv axis then V = (0, 0, Vz) and V . N = Vz . C So we have to consider the sign of C, the z component of the normal vector N. It is shown in Figure.5.14. In a right-handed viewing system with viewing direction along the negative Zv axis (Figure.5.15) the polygon is back-face if C < 0. If C = 0, we cannot see any face, since the viewing direction is graying that polygon. Thus we can label any polygon as a back-face if its normal vector has a Z component value C ≤ 0 Back-faces have normal vectors that point away from the viewing position and are identified by C ³ 0 when the viewing direction is along the positive Zv axis.
- 10. DEPTH BUFFER METHOD (OR) Z- BUFFER ALGORITHM This method compares surface depths at each pixel position on the projection plane. The surface depth is measured from the view plane along the Z axis of a viewing system. When object description is converted to projection coordinates (x, y, z), each pixel position on the view plane is specified by x and y coordinate, and z value gives the depth information. Thus object depths can be compared by comparing the z-values. The Z buffer algorithm is usually implemented in the normalized coordinates, so that z values range from 0 at the back clipping plane to 1 at the front clipping plane. The implementation requires another buffer memory called z-buffer along with the frame buffer memory required for raster display devices. A z-buffer is used to store depth values for each (x, y) position as surfaces are processed and the frame buffer stores the intensity values for each position. At the beginning, z-buffer is initialized to 0, representing the z-value at the back clipping plane and the frame buffer is initialized to the background color. Each surface listed in the display file is then processed one scan line at a time, calculating the depth (z value) at each (x, y) pixel position. The calculated depth value is compared to the value previously stored in the z-buffer at that position. If the calculated depth value is greater than the value stored in the z-buffer, the new depth value is stored, and the surface intensity at that position is determined and placed in the same x,y location in the frame buffer. For example in the following Fig.5.16 among three surfaces, surface s has the smallest depth at view position (x, y) and hence highest z value. So it is visible at that position.
- 11. Z-buffer Algorithm Step-1: Initialize the z-buffer and frame buffer so that for all buffer positions Z-buffer (x, y) = 0 and frame buffer(x, y) = Ibackground Step-2: During scan conversion process, for each position on each polygon surface, compare depth values to previously stored values in the depth buffer to determine visibility. Calculate z-value for each (x,y) position on the polygon. Advantages (1) It is very easy to implement. (2) It can be implemented in hardware to overcome the speed problem. (3) Since the algorithm processes objects one at a time, the total number of polygons in a picture can be arbitrarily large. If z>Z-buffer(x,y) then set Step-3: Stop After processing all the surfaces, the z-buffer, contains depth values for the variable surfaces and the frame buffer contains the corresponding intensity values for those surfaces. To calculate z-values, the plane equation Ax + By + Cz + D =0 is used where (x, y, z) is any point on the plane, and the coefficient A, B, C and D are constants describing the spatial properties of the plane. If at (x,y), the above equation evaluates to z, then at (x+Dx, y) the value of z is Only one subtraction is needed to calculate z(x+1, y), given z(x, y) since the quotient A/C is constant and Δx=1. A similar incremental calculation can be performed to determine the first value of z on the next scanline, decrementing by B/C for each Δy.
- 12. Disadvantages (1) It requires an additional buffer and hence the large memory. (2) It is a time consuming process as it requires comparison for each pixel instead of the entire polygon. BUFFER METHOD An extension of the ideas in the depth buffer method is the A-buffer method. A-buffer method represents an antialiased, area-averaged, accumulation-buffer method. It expands the depth buffer so that each position in the buffer can reference a linked list of surfaces. More than one surface intensity can be taken into consideration at each pixel position and object edges can be antialiased. Each position in the A-buffer has 2 fields. (1) Depth field-stores a positive or negative real number. (2) Intensity field-stores surface-intensity information or a pointer value. If the depth field is positive, the number stored at that position in the depth of a single surface overlapping the corresponding pixel area. The intensity field stores the RGB components of the surface color at that point and the percent of pixel coverage. If the depth field is negative, it indicates multiple surface contributions to the pixel intensity. The intensity field stores a pointer to a linked list of surface data. The data for each surface in the linked list includes : 1. RGB intensity components. 2. Opacity parameter. 3. Depth. 4. Percent of area coverage. 5. Surface identifier. 6. Other surface rendering parameters. 7. Pointer to next surface. The A-buffer can be constructed using methods similar to those in the depth-buffer algorithm. Scan lines are processed to determine surface overlaps of pixels across the individual scan lines. Surfaces are subdivided into a polygon mesh and clipped against the pixel boundaries. Using the opacity factors and percent of surface overlaps, the intensity of each pixel is calculated as an average of the contributions from the overlapping surfaces.
- 13. SCANLINE METHOD A Scan line method of hidden surface removal is an another approach of image space method. This method deals with more than one surface. As each scanline is processed, it examines all polygon surfaces intersecting that line to determine which are visible. It performs the depth calculation and finds which polygon is nearest to the view plane. For the positions along this scan line between edges AD and BC, only the flag for surface S1 is ON. Therefore, no depth calculations are necessary and intensity information for flag for surface S2 is ON and during that position of scan line the intensity information for surface S2 is entered into Finally, it enters the intensity value of the nearest polygon at that position into the frame buffer. The scanline algorithm maintains active edge list. The active edge list contains only edges that cross the current scan line, sorted in order ofincreasing x. scanline method of hidden surface removal also stores a flag for each surface that is set ONor OFF to indicate whether a position along a scan line is inside or outside of the surface. Scanlines are processed from left to right.ftmost boundary of the surface, the surface flag is turned ON, and at the rightmostboundary, it is turned OFF. The Figure.5.17 illustrates the scanline method for hidden surface removal. Here, the active edge list for scanline 1 contains the information for edges AD, BC, EH, and FG.
- 14. the frame buffer. For scan line 2, the active edge list contains edges AD, EH, BC and FG. Along the scan line 2from edge AD to edge EH, only the flag for surface S1 is ON. Between edges EH and BC, the flags for both surfaces are ON. In this portion of scanline2, thedepth calculations are necessary. Here we have assumed that the depth of S1 is less than the depth of S2 and hence the intensitiesof surface S1