IGNOU MMTE 4 2026 - Transform the scene from the world coordinate system to the viewing co

Question

Transform the scene from the world coordinate system to the viewing coordinate system with viewpoint at (2,-1,3). The view plane normal vector is (3,-2,4) and the view-up vector is (0,1,2).
b) An object is rotated about the z-axis with an angle of 30° and then it is uniformly scaled up by a factor of 4. Find the resultant transformation matrix.

06 Mar 2026

Answer :

Word Count : 645

To transform the scene from the world coordinate system (WCS) to the viewing coordinate system (VCS), we follow the standard procedure of constructing a viewing transformation matrix. Part (a): WCS → VCS Transformation Given: * Viewpoint ( V = (2, -1, 3) ) * View plane normal ( \mathbf{n} = (3, -2, 4) ) * View-up vector ( \mathbf{v} = (0, 1, 2) ) Step 1: Compute the orthonormal basis vectors for the viewing coordinate system. * n-axis (view direction): unit vector along the view plane normal [ \mathbf{n_{unit}} = \frac{\mathbf{n}}{|\mathbf{n}|}, \quad |\mathbf{n}| = \sqrt{3^2 + (-2)^2 + 4^2} = \sqrt{9+4+16} = \sqrt{29} ] [ \mathbf{n_{unit}} = \left( \frac{3}{\sqrt{29}}, \frac{-2}{\sqrt{29}}, \frac{4}{\sqrt{29}} \right) ] * u-axis (perpendicular to n and view-up v): [ \mathbf{u} = \mathbf{v} \times \mathbf{n} ] Compute cross product: [ \mathbf{v} \times \mathbf{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & 1 & 2 \ 3 & -2 & 4 \end{vmatrix} = \mathbf{i}(1*4 - 2*(-2)) - \mathbf{j}(0*4 - 2*3) + \mathbf{k}(0*(-2) - 1*3) ] [ = \mathbf{i}(4 +4) - \mathbf{j}(0-6)

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