Past exam of the computer science course of the University of Oxford / 2015 / Computer graphics / 2 / b / i / Solution

In the solid-modelling convention used here, a rigid-body transformation is an orientation-preserving rigid transformation, represented by a proper rotation and a translation. The proposed upper-left block $S$ is orthogonal but hasIt is a reflection in the plane $x=y$. It preserves distances, but reverses handedness, so it cannot describe continuous motion of a rigid solid through proper rotations and translations. Some mathematical uses of rigid transformation include this orientation-reversing case; the determinant is the distinction relevant to the warning.

In a solid modeller, triangle winding and outward normal vectors must agree. A reflection reverses the winding: for edge vectors $a,b$,Simply transforming stored normal vectors by $S$ while retaining the original vertex order therefore makes the geometric and stored normals disagree. Back-face culling, inside/outside tests and Phong reflection model calculations can then use the wrong side. Code assuming a positive oriented volume can likewise interpret the reflected solid incorrectly. A modeller can support reflections if it deliberately corrects these orientation conventions.

A suitable alternative rotates by $-\pi /2$ about $z$ and then translates upward by height. Writing $h=\text{height}$,Its rotation matrix has determinant $+1$. The source bounds $0\le x\le h$, $0\le y\le w$, $z=0$ become $0\le x^{\prime }\le w$, $0\le y^{\prime }\le h$, $z^{\prime }=0$, where $w=\text{width}$. A horizontal stripe $y=k\text{step}$ becomes the vertical stripe $x^{\prime }=k\text{step}$, so even the stripe positions from the left edge are preserved. The face's $+z$ normal vector is also preserved.