Questions and Answers

This is a place where everyone can ask questions about things he/she did not understand or which need clarification.

Kann mir jemand die Cross ratios erklären? Insbesondere was Prof. Kobbelt damit ausdrücken will? Danke, Verena

(Can somebody explain the cross ratios to me? Especially what Prof. Kobbelt wants to say about it? Thanks, Verena)

If you have three points on a line, A, B and C, you can express C in terms of A and B: $C=\alpha A + (1-\alpha) B$.

This even holds when you apply a linear or an affine mapping to these points, due to linearity and additivity: $f(C)=f(\alpha A + (1-\alpha) B)=\alpha f(A) + (1-\alpha) f(B)$.

However, it does not hold in case of projective maps. There is a good picture visualizing this effect in the CG1 script, page 38 (Figure 3.2). The ratio $\alpha$ in world coordinates is about 0.5, while on the image plane the point is shifted towards B. Cross ratios are used to establish a relationship between world space ratios and image space ratios.

This is not an issue in our rendering pipeline since lines are still mapped to lines after projective transformations and we do not require world-space ratios projected to the image plane anywhere in the pipeline as far as I can remember. It would, however, be an issue for some operations - and in fact it does become when talking about the triangle rendering. Two examples would be the normal interpolation in world space rather than image space, and the perspectively corrected texture coordinate computation.

Meine Frage geht noch mehr in die Details:

Erstens: gibt es einen Grund (bzw. wurde er genannt), dass die projektiven Transformationen invariant gegenüber den cross ratios sind?

Zweitens: Wieso wird $CR(0, \alpha, 1, \infty)$ berechnet, und wie folgert man daraus, dass $(0, \alpha, 1, \infty)$ auf $(0, \beta, 1, \gamma)$ abgebildet wird (Folien 131 und 132)? Nochmals danke, Verena

(My question is somewhat more specific: First, why are the projective transformations invariant to the cross ratios? And second, why is $CR(0, \alpha, 1, \infty)$ computed and how does that explain the mapping from $(0, \alpha, 1, \infty)$ to $(0, \beta, 1, \gamma)$ (see slides 131 and 132)? Thanks, Verena)

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