# Transformations and change of basis in CG

Sometimes in CG literature a transformation is equated to a coordinate system change. This is fine as long as the transformation includes only a rotation and a translation. When scaling is involved, however, this seems like an incorrect interpretation. The problem is specifically with how lengths and angles are measured after the transformation. Angles and length are invariant to coordinate system change: $$\langle a, b \rangle = \sum_{i,j}g_{ij}a^ib^j$$. In graphics, this is often substituted with the expression valid only for orthonormal bases: $$\langle a, b \rangle = \sum_{i}a_ib^i$$, regardless of the supposed basis that $$a,b$$ are in (the index of $$a$$ is lowered as if the basis was orthonormal without involving the metric tensor). So while this makes sense as a transformation, I do not see how this is reconciled with the idea of a change of basis. Am I missing something, or is the interpretation wrong? (think of DX/GL's standard transformations between spaces in the rendering pipeline)

• Correct me if I am wrong, but that means that the correct interpretation is that the models live in world space to begin with and are instead transformed ("deformed") with the modelToWorld matrix, instead of actually changing coordinate systems. As far as the normals go, I have no issue with that - they can be interpreted as covectors (up to a multiplicative factor) and thus do not require the metric tensor to lower the index. Jun 27, 2020 at 13:14
• Coordinate system change implies that the metric tensor changes. If we go from a basis A to a basis B with a map F, then the metric tensor changes as: $g_{ij}' = \sum_{k,l}F^k_iF^l_jg_{kl}$, which affects how the inner product is computed in the new basis. Instead if we do not interpret this as acting on the basis, but rather only on the components of vectors (that is, the basis is preserved but points are translated, rotated, scaled), the inner product does not change (which seems to be the approach most often considered in CG). Jun 27, 2020 at 16:28