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.
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 ________ ___ __________ ____ ______ ______ ______ _____ _________ _________ __________ ___.
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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.
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