<p>I think the terminology makes it seem more complicated than it is. The &quot;Texture Space Jacobean&quot; Matrix in your slide here is just a Linear Transformation Matrix.</p>

CC BY-SA 4.0
2022-02-05T17:15:24.400Z
I don't think this is inverse = transpose, it's the inverse of the transpose/transpose of the inverse. Let $d(u(x,y), v(x,y))$ then $d_x = d_u u_x + d_v v_x$ and $d_y = d_u u_y + d_v v_y$. That is $\begin{bmatrix}d_x \\ d_y \end{bmatrix} = \begin{bmatrix} u_x & u_y \\ v_x & v_y\end{bmatrix}^T\begin{bmatrix} d_u \\ d_v \end{bmatrix}$. Now multiply both sides by the inverse transpose and you get what they have. Also the transpose doesn't have to be equal to the inverse. That's true only for orthogonal matrices. This doesn't have to be the case for screen to uv mappings.
lightxbulb
CC BY-SA 4.0
2022-02-06T00:38:46.253Z
Now the *-T* makes sense, too.
luser droog
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2022-01-31T19:11:29.280Z
2022-02-05T23:38:08.530Z
2022-02-05T23:38:08.530Z
luser droog
<p><a href="https://developer.amd.com/wordpress/media/2012/10/Isidoro-ShadowMapping.pdf" rel="nofollow noreferrer">This source</a> on slide 38 on receiver plane depth bias I can find compute a matrix (Jacobian of screen space uv derivatives) which is used to transform the screen space depth derivatives from screen space to texture space.</p>
<p><a href="https://i.sstatic.net/JNOHd.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JNOHd.png" alt="enter image description here" /></a></p>
<p>But why does this work? I'm not familiar with the usage of Jacobian matrices and how they relate to coordinate system transformations. Does the basic Jacobian shown in the slide transform from texture space to screen space, or why does it have to be inverse-transposed (I suppose of the former was true it could just be inverted?).</p>

CC BY-SA 4.0
2022-02-05T17:18:52.577Z
See my comment on the answer below. It's just the chain rule in matrix form.
lightxbulb
<transformations><shadow-mapping>
Receiver plane depth bias - matrix explanation / derivation
cg_guy420
# Receiver plane depth bias - matrix explanation / derivation

(Asked by: username_0 on 2022-01-31)

[This source](https://developer.amd.com/wordpress/media/2012/10/Isidoro-ShadowMapping.pdf) on slide 38 on receiver plane depth bias I can find compute a matrix (Jacobian of screen space uv derivatives) which is used to transform the screen space depth derivatives from screen space to texture space.

[![enter image description here](https://i.sstatic.net/JNOHd.png)](https://i.sstatic.net/JNOHd.png)

But why does this work? I'm not familiar with the usage of Jacobian matrices and how they relate to coordinate system transformations. Does the basic Jacobian shown in the slide transform from texture space to screen space, or why does it have to be inverse-transposed (I suppose of the former was true it could just be inverted?).

**Tags:** transformations, shadow-mapping

> **Comments:**
> **username_1**: See my comment on the answer below. It's just the chain rule in matrix form.  

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## 1 answers

### Answer by username_2 on 2022-01-31. Score: 0

I think the terminology makes it seem more complicated than it is. The "Texture Space Jacobean" Matrix in your slide here is just a Linear Transformation Matrix.

> **Comments:**
> **username_1**: I don't think this is inverse = transpose, it's the inverse of the transpose/transpose of the inverse. Let $d(u(x,y), v(x,y))$ then $d_x = d_u u_x + d_v v_x$ and $d_y = d_u u_y + d_v v_y$. That is $\begin{bmatrix}d_x \\ d_y \end{bmatrix} = \begin{bmatrix} u_x & u_y \\ v_x & v_y\end{bmatrix}^T\begin{bmatrix} d_u \\ d_v \end{bmatrix}$. Now multiply both sides by the inverse transpose and you get what they have. Also the transpose doesn't have to be equal to the inverse. That's true only for orthogonal matrices. This doesn't have to be the case for screen to uv mappings.  
> **username_2**: Now the *-T* makes sense, too.  

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