Window to Viewport Transformation Matrix
The transformation from a 3D window to a viewport involves scaling and translation. We need to find the matrix that maps the window coordinates to the viewport coordinates.
1. Calculate Scaling Factors
The scaling factor for each axis is the ratio of the viewport dimension to the window dimension.
- X-axis: Window range = $4 - 1 = 3$. Viewport range = $8 - 2 = 6$. Scaling factor $Sx = \frac{6}{3} = 2$.
- Y-axis: Window range = $5 - 1 = 4$. Viewport range = $10 - 4 = 6$. Scaling factor $Sy = \frac{6}{4} = \frac{3}{2}$.
- Z-axis: Window range = $6 - 1 = 5$. Viewport range = $8 - 2 = 6$. Scaling factor $Sz = \frac{6}{5}$.
2. Determine Translation Components
The transformation can be represented as mapping the window's minimum coordinate to the viewport's minimum coordinate, followed by scaling. The general form of the transformation for a point $(x_w, y_w, z_w)$ to $(x_{vp}, y_{vp}, z_{vp})$ is:
$x_{vp} = Sx \cdot (x_w - x_{w_{min}}) + x_{vp_{min}}$ $y_{vp} = Sy \cdot (y_w - y_{w_{min}}) + y_{vp_{min}}$ $z_{vp} = Sz \cdot (z_w - z_{w_{min}}) + z_{vp_{min}}$Substituting the values:
- $x_{vp} = 2 \cdot (x_w - 1) + 2 = 2x_w - 2 + 2 = 2x_w$
- $y_{vp} = \frac{3}{2} \cdot (y_w - 1) + 4 = \frac{3}{2}y_w - \frac{3}{2} + 4 = \frac{3}{2}y_w + \frac{5}{2}$
- $z_{vp} = \frac{6}{5} \cdot (z_w - 1) + 2 = \frac{6}{5}z_w - \frac{6}{5} + 2 = \frac{6}{5}z_w + \frac{4}{5}$
3. Construct the Transformation Matrix
The transformation can be represented using homogeneous coordinates and a 4x4 matrix. The derived equations correspond to the following matrix:
$ M = \begin{bmatrix} Sx & 0 & 0 & Tx \\ 0 & Sy & 0 & Ty \\ 0 & 0 & Sz & Tz \\ 0 & 0 & 0 & 1 \end{bmatrix} $Where:
- $Tx = -Sx \cdot x_{w_{min}} + x_{vp_{min}} = -2 \cdot 1 + 2 = 0$
- $Ty = -Sy \cdot y_{w_{min}} + y_{vp_{min}} = -\frac{3}{2} \cdot 1 + 4 = \frac{5}{2}$
- $Tz = -Sz \cdot z_{w_{min}} + z_{vp_{min}} = -\frac{6}{5} \cdot 1 + 2 = \frac{4}{5}$
Therefore, the final normalized matrix is:
This matches Option A.
Important Questions from Three Dimensional Concepts
Which of the following statements are true.
A. Painters' algorithm is also called depth sort algorithm.
B. Gourad shading is on interpolating shading method.
C. Gourad shading is an averaging shading method.
D. The Z-buffer algorithm finds the largest depth value Z.
Choose the correct answer from the options given below:
In the context of 3D Computer graphics, which of the following statements is/are correct?
(A) Under perspective projection, each set of parallel lines in the object do not stay parallel in the image (except those that are parallel to the viewplane to start with).
(B) Applying a perspective transformation in the graphics pipeline to a vertex involves dividing by its 'z' coordinate.
(C) Perspective transformation is a linear transformation.
Choose the correct answer from the options given below:
In 3D graphics, which statements are true?
P: Back-face culling is an example of image-precision visible-surface determination.
Q: In single-sample rasterization, a Z-buffer stores a depth value for each pixel and uses depth comparisons to determine visible surfaces; its numerical precision depends on the implementation.
In 3D Graphics, which of the following statements about perspective and parallel projection is/are true?
P: In a perspective projection, the farthest an object is from the center of projection, the smaller it appears.
Q: Parallel projection is equivalent to a perspective projection where the viewer is standing infinitely far away
R: Perspective projections do not preserve straight lines.Which of the following statements is/are True regarding the solution to the visibility problem in 3D graphics?
S1: The Painter’s algorithm sorts polygons by depth and then paints (scan - converts) each Polygon on to the screen starting with the most nearest polygon.
S2: Backface Culling refers to eliminating geometry with backfacing normals.
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