diff options
| author |  Yuqian Li <liyuqian@google.com> | 2018-01-02 16:17:11 -0500 |
|---|---|---|
| committer |  Skia Commit-Bot <skia-commit-bot@chromium.org> | 2018-01-02 21:20:11 +0000 |
| commit | c89ac122ef3ec4dd23f33ef5384e9126c923d7c2 (patch) | |
| tree | 2355fc40f9aa582e14cb0a13b9cb1e365c372103 /site/dev | |
| parent | 632156d835940df95a900d8e8c9851eda0cb917c (diff) | |
Document the 2pt conical gradient
I've also updated/simplified our source code
to match the document.
Please ignore everything except the md and cpp file.
See the rendered document in:
https://skia.org/dev/design/conical?cl=89340
If the math is not rendered properly, refresh the page.
No-Try: true
Docs-Preview: https://skia.org/?cl=89340
Bug: skia:
Change-Id: I9b9306c1979960ccec0d3ab833391c649edb833c
Reviewed-on: https://skia-review.googlesource.com/89340
Commit-Queue: Yuqian Li <liyuqian@google.com>
Reviewed-by: Greg Daniel <egdaniel@google.com>
Diffstat (limited to 'site/dev')
21 files changed, 2757 insertions, 0 deletions
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src='https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-MML-AM_CHTML'></script> + +We present a fast shading algorithm (compared to bruteforcely solving the quadratic equation of +gradient $t$) for computing the two-point conical gradient (i.e., createRadialGradient in +[spec](https://html.spec.whatwg.org/multipage/canvas.html#dom-context-2d-createradialgradient)). + +This document has 3 parts: + +1. Problem Statement and Setup +2. Algorithm +3. Appendix + +Part 1 and 2 are self-explanatory. Part 3 shows how to geometrically proves our Theorem 1 in part +2; it's more complicated but it gives us a nice picture about what's going on. + +## Problem Statement and Setup + +Let two circles be $C_0, r_0$ and $C_1, r_1$ where $C$ is the center and $r$ is the radius. For any +point $P = (x, y)$ we want the shader to quickly compute a gradient $t \in \mathbb R$ such that $p$ +is on the linearly interpolated circle with center $C_t = (1-t) \cdot C_0 + t \cdot C_1$ and radius +$r_t = (1-t) \cdot r_0 + t \cdot r_1 > 0$ (note that radius $r_t$ has to be *positive*). If +there are multiple (at most 2) solutions of $t$, choose the bigger one. + +There are two degenerated cases: + +1. $C_0 = C_1$ so the gradient is essentially a simple radial gradient. +2. $r_0 = r_1$ so the gradient is a single strip with bandwidth $2 r_0 = 2 r_1$. + +<!-- TODO maybe add some fiddle or images here to illustrate the two degenerated cases --> + +They are easy to handle so we won't cover them here. From now on, we assume $C_0 \neq C_1$ and $r_0 +\neq r_1$. + +As $r_0 \neq r_1$, we can find a focal point $C_f = (1-f) \cdot C_0 + f \cdot C_1$ where its +corresponding linearly interpolated radius $r_f = (1-f) \cdot r_0 + f \cdot r_1 = 0$. +Solving the latter equation gets us $f = r_0 / (r_0 - r_1)$. + +As $C_0 \neq C_1$, focal point $C_f$ is different from $C_1$ unless $r_1 = 0$. If $r_1 = 0$, we can +swap $C_0, r_0$ with $C_1, r_1$, compute swapped gradient $t_s$ as if $r_1 \neq 0$, and finally set +$t = 1 - t_s$. The only catch here is that with multiple solutions of $t_s$, we shall choose the +smaller one (so $t$ could be the bigger one). + +Assuming that we've done swapping if necessary so $C_1 \neq C_f$, we can then do a linear +transformation to map $C_f, C_1$ to $(0, 0), (1, 0)$. After the transformation: + +1. All centers $C_t = (x_t, 0)$ must be on the $x$ axis +2. The radius $r_t$ is $x_t r_1$. +3. Given $x_t$ , we can derive $t = f + (1 - f) x_t$ + +From now on, we'll focus on how to quickly computes $x_t$. Note that $r_t > 0$ so we're only +interested positive solution $x_t$. Again, if there are multiple $x_t$ solutions, we may want to +find the bigger one if $1 - f > 0$, and smaller one if $1 - f < 0$, so the corresponding $t$ is +always the bigger one (note that $f \neq 1$, otherwise we'll swap $c_0, r_0$ with $c_1, r_1$). + +## Algorithm + +**Theorem 1.** The solution to $x_t$ is + +1. $\frac{x^2 + y^2}{(1 + r_1) x} = \frac{x^2 + y^2}{2 x}$ if $r_1 = 1$ +2. $\left(\sqrt{(r_1^2 - 1) y ^2 + r_1^2 x^2} - x\right) / (r_1^2 - 1)$ if $r_1 > 1$ +3. $\left(\pm \sqrt{(r_1^2 - 1) y ^2 + r_1^2 x^2} - x\right) / (r_1^2 - 1)$ if $r_1 < 1$. + +Case 2 always produces a valid $x_t$. Case 1 and 3 requires $x > 0$ to produce valid $x_t > 0$. Case +3 may have no solution at all if $(r_1^2 - 1) y^2 + r_1^2 x^2 < 0$. + +*Proof.* Algebriacally, solving the quadratic equation $(x_t - x)^2 + y^2 = (x_t r_1)^2$ and +eliminate negative $x_t$ solutions get us the theorem. + +Alternatively, we can also combine Corollary 2., 3., and Lemma 4. in the Appendix to geometrically +prove the theorem. $\square$ + +Theorem 1 by itself is not sufficient for our shader algorithm because: + +1. we still need to compute $t$ from $x_t$ (remember that $t = f + (1-f) x_t$); +2. we still need to handle cases of choosing the bigger/smaller $x_t$; +3. we still need to handle the swapped case (we swap $C_0, r_0$ with $C_1, r_1$ if $r_1 = 0$); +4. there are way too many multiplications and divisions in Theorem 1 that would slow our shader. + +Issue 2 and 3 are solved by generating different shader code based on different situations. So they +are mainly correctness issues rather than performance issues. Issue 1 and 4 are performance +critical, and they will affect how we handle issue 2 and 3. + +The key to handle 1 and 4 efficiently is to fold as many multiplications and divisions into the +linear transformation matrix, which the shader has to do anyway (remember our linear transformation +to map $C_f, C_1$ to $(0, 0), (1, 0)$). + +For example, let $\hat x, \hat y = |1-f|x, |1-f|y$. Computing $\hat x_t$ with respect to $\hat x, +\hat y$ allow us to have $t = f + (1 - f)x_t = f + \text{sign}(1-f) \cdot \hat x_t$. That saves us +one multiplication. Applying similar techniques to Theorem 1 gets us: + +1. If $r_1 = 1$, let $x' = x/2,~ y' = y/2$, then $x_t = (x'^2 + y'^2) / x'$. +2. If $r_1 > 1$, let $x' = r_1 / (r_1^2 - 1) x,~ y' = \frac{\sqrt{r_1^2 - 1}}{r_1^2 - 1} y$, then + $x_t = \sqrt{x'^2 + y'^2} - x' / r_1$ +3. If $r_1 < 1$, let $x' = r_1 / (r_1^2 - 1) x,~ y' = \frac{\sqrt{1 - r_1^2}}{r_1^2 - 1} y$, then + $x_t = \pm\sqrt{x'^2 - y'^2} - x' / r_1$ + +Combining it with the swapping, the equation $t = f + (1-f) x_t$, and the fact that we only want +positive $x_t > 0$ and bigger $t$, we have our final algorithm: + +**Algorithm 1.** + +1. Let $C'_0, r'_0, C'_1, r'_1 = C_0, r_0, C_1, r_1$ if there is no swapping and $C'_0, + r'_0, C'_1, r'_1 = C_1, r_1, C_0, r_0$ if there is swapping. +2. Let $f = r'_0 / (r'_0 - r'_1)$ and $1 - f = r'_1 / (r'_1 - r'_0)$ +3. Let $x' = x/2,~ y' = y/2$ if $r_1 = 1$, and + $x' = r_1 / (r_1^2 - 1) x,~ y' = \sqrt{|r_1^2 - 1|} / (r_1^2 - 1) y$ if $r_1 \neq 1$ +4. Let $\hat x = |1 - f|x', \hat y = |1 - f|y'$ +5. If $r_1 = 1$, let $\hat x_t = (\hat x^2 + \hat y^2) / \hat x$ +6. If $r_1 > 1$, + let $\hat x_t = \sqrt{\hat x^2 + \hat y^2} - \hat x / r_1$ +7. If $r_1 < 1$ + 1. return invalid if $\hat x^2 - \hat y^2 < 0$ + 2. let $\hat x_t = -\sqrt{\hat x^2 - \hat y^2} - \hat x / r_1$ if we've swapped $r_0, r_1$, + or if $1 - f < 0$ + + 3. let $\hat x_t = \sqrt{\hat x^2 - \hat y^2} - \hat x / r_1$ otherwise + +8. $t$ is invalid if $\hat x_t < 0$ (this check is unnecessary if $r_1 > 1$) +9. Let $t = f + \text{sign}(1 - f) \hat x_t$ +10. If swapped, let $t = 1 - t$ + +In step 7, we try to select either the smaller or bigger $\hat x_t$ based on whether the final $t$ +has a negative or positive relationship with $\hat x_t$. It's negative if we've swapped, or if +$\text{sign}(1 - f)$ is negative (these two cannot both happen). + +Note that all the computations and if decisions not involving $\hat x, \hat y$ can be precomputed +before the shading stage. The two if decisions $\hat x^2 - \hat y^2 < 0$ and $\hat x^t < 0$ can +also be omitted by precomputing the shading area that never violates those conditions. + +The number of operations per shading is thus: + +* 1 addition, 2 multiplications, and 1 division if $r_1 = 1$ +* 2 additions, 3 multiplications, and 1 sqrt for $r_1 \neq 1$ (count substraction as addition; + dividing $r_1$ is multiplying $1/r_1$) +* 1 more addition operation if $f \neq 0$ +* 1 more addition operation if swapped. + +In comparison, for $r_1 \neq 1$ case, our current raster pipeline shading algorithm (which shall +hopefully soon be upgraded to the algorithm described here) mainly uses formula $$t = 0.5 \cdot +(1/a) \cdot \left(-b \pm \sqrt{b^2 - 4ac}\right)$$ It precomputes $a = 1 - (r_1 - r_0)^2, 1/a, r1 - +r0$. Number $b = -2 \cdot (x + (r1 - r0) \cdot r0)$ costs 2 multiplications and 1 addition. Number +$c = x^2 + y^2 - r_0^2$ costs 3 multiplications and 2 additions. And the final $t$ costs 5 more +multiplications, 1 more sqrt, and 2 more additions. That's a total of 5 additions, 10 +multiplications, and 1 sqrt. (Our algorithm has 2-4 additions, 3 multiplications, and 1 sqrt.) Even +if it saves the $0.5 \cdot (1/a), 4a, r_0^2$ and $(r_1 - r_0) r_0$ multiplications, there are still +6 multiplications. Moreover, it sends in 4 unitofmrs to the shader while our algorithm only needs 2 +uniforms ($1/r_1$ and $f$). + +## Appendix + +**Lemma 1.** Draw a ray from $C_f = (0, 0)$ to $P = (x, y)$. For every +intersection points $P_1$ between that ray and circle $C_1 = (1, 0), r_1$, there exists an $x_t$ +that equals to the length of segment $C_f P$ over length of segment $C_f P_1$. That is, +$x_t = || C_f P || / ||C_f P_1||$ + +*Proof.* Draw a line from $P$ that's parallel to $C_1 P_1$. Let it intersect with $x$-axis on point +$C = (x', y')$. + +<img src="conical/lemma1.svg"/> + +Triangle $\triangle C_f C P$ is similar to triangle $\triangle C_f C_1 P_1$. +Therefore $||P C|| = ||P_1 C_1|| \cdot (||C_f C|| / ||C_f C_1||) = r_1 x'$. Thus $x'$ is a solution +to $x_t$. Because triangle $\triangle C_f C P$ and triangle $\triangle C_f C_1 P_1$ are similar, $x' += ||C_f C_1|| \cdot (||C_f P|| / ||C_f P_1||) = ||C_f P|| / ||C_f P_1||$. $\square$ + +**Lemma 2.** For every solution $x_t$, if we extend/shrink segment $C_f P$ to $C_f P_1$ with ratio +$1 / x_t$ (i.e., find $P_1$ on ray $C_f P$ such that $||C_f P_1|| / ||C_f P|| = 1 / x_t$), then +$P_1$ must be on circle $C_1, r_1$. + +*Proof.* Let $C_t = (x_t, 0)$. Triangle $\triangle C_f C_t P$ is similar to $C_f C_1 P_1$. Therefore +$||C_1 P_1|| = r_1$ and $P_1$ is on circle $C_1, r_1$. $\square$ + +**Corollary 1.** By lemma 1. and 2., we conclude that the number of solutions $x_t$ is equal to the +number of intersections between ray $C_f P$ and circle $C_1, r_1$. Therefore + +* when $r_1 > 1$, there's always one unique intersection/solution; we call this "well-behaved"; this + was previously known as the "inside" case; +* when $r_1 = 1$, there's either one or zero intersection/solution (excluding $C_f$ which is always + on the circle); we call this "focal-on-circle"; this was previously known as the "edge" case; + +<img src="conical/corollary2.2.1.svg"/> +<img src="conical/corollary2.2.2.svg"/> + +* when $r_1 < 1$, there may be $0, 1$, or $2$ solutions; this was also previously as the "outside" + case. + +<img src="conical/corollary2.3.1.svg" width="30%"/> +<img src="conical/corollary2.3.2.svg" width="30%"/> +<img src="conical/corollary2.3.3.svg" width="30%"/> + +**Lemma 3.** When solution exists, one such solution is +$$ + x_t = {|| C_f P || \over ||C_f P_1||} = \frac{x^2 + y^2}{x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}} +$$ + +*Proof.* As $C_f = (0, 0), P = (x, y)$, we have $||C_f P|| = \sqrt(x^2 + y^2)$. So we'll mainly +focus on how to compute $||C_f P_1||$. + +**When $x \geq 0$:** + +<img src="conical/lemma3.1.svg"/> + +Let $X_P = (x, 0)$ and $H$ be a point on $C_f P_1$ such that $C_1 H$ is perpendicular to $C_1 +P_1$. Triangle $\triangle C_1 H C_f$ is similar to triangle $\triangle P X_P C_f$. Thus +$$||C_f H|| = ||C_f C_1|| \cdot (||C_f X_P|| / ||C_f P||) = x / \sqrt{x^2 + y^2}$$ +$$||C_1 H|| = ||C_f C_1|| \cdot (||P X_P|| / ||C_f P||) = y / \sqrt{x^2 + y^2}$$ + +Triangle $\triangle C_1 H P_1$ is a right triangle with hypotenuse $r_1$. Hence +$$ ||H P_1|| = \sqrt{r_1^2 - ||C_1 H||^2} = \sqrt{r_1^2 - y^2 / (x^2 + y^2)} $$ + +We have +\begin{align} + ||C_f P_1|| &= ||C_f H|| + ||H P_1|| \\\\\\ + &= x / \sqrt{x^2 + y^2} + \sqrt{r_1^2 - y^2 / (x^2 + y^2)} \\\\\\ + &= \frac{x + \sqrt{r_1^2 (x^2 + y^2) - y^2}}{\sqrt{x^2 + y^2}} \\\\\\ + &= \frac{x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}}{\sqrt{x^2 + y^2}} +\end{align} + +**When $x < 0$:** + +Define $X_P$ and $H$ similarly as before except that now $H$ is on ray $P_1 C_f$ instead of +$C_f P_1$. + +<img src="conical/lemma3.2.svg"/> + +As before, triangle $\triangle C_1 H C_f$ is similar to triangle $\triangle P X_P C_f$, and triangle +$\triangle C_1 H P_1$ is a right triangle, so we have +$$||C_f H|| = ||C_f C_1|| \cdot (||C_f X_P|| / ||C_f P||) = -x / \sqrt{x^2 + y^2}$$ +$$||C_1 H|| = ||C_f C_1|| \cdot (||P X_P|| / ||C_f P||) = y / \sqrt{x^2 + y^2}$$ +$$ ||H P_1|| = \sqrt{r_1^2 - ||C_1 H||^2} = \sqrt{r_1^2 - y^2 / (x^2 + y^2)} $$ + +Note that the only difference is changing $x$ to $-x$ because $x$ is negative. + +Also note that now $||C_f P_1|| = -||C_f H|| + ||H P_1||$ and we have $-||C_f H||$ instead of +$||C_f H||$. That negation cancels out the negation of $-x$ so we get the same equation +of $||C_f P_1||$ for both $x \geq 0$ and $x < 0$ cases: + +$$ + ||C_f P_1|| = \frac{x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}}{\sqrt{x^2 + y^2}} +$$ + +Finally +$$ + x_t = \frac{||C_f P||}{||C_f P_1||} = \frac{\sqrt{x^2 + y^2}}{||C_f P_1||} + = \frac{x^2 + y^2}{x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}} +$$ $\square$ + +**Corollary 2.** If $r_1 = 1$, then the solution $x_t = \frac{x^2 + y^2}{(1 + r_1) x}$, and +it's valide (i.e., $x_t > 0$) iff $x > 0$. + +*Proof.* Simply plug $r_1 = 1$ into the formula of Lemma 3. $\square$ + +**Corollary 3.** If $r_1 > 1$, then the unique solution is +$x_t = \left(\sqrt{(r_1^2 - 1) y ^2 + r_1^2 x^2} - x\right) / (r_1^2 - 1)$. + +*Proof.* From Lemma 3., we have + +\begin{align} + x_t &= \frac{x^2 + y^2}{x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}} \\\\\\ + &= { + (x^2 + y^2) \left ( -x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2} \right ) + \over + \left (x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2} \right ) + \left (-x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2} \right ) + } \\\\\\ + &= { + (x^2 + y^2) \left ( -x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2} \right ) + \over + -x^2 + (r_1^2 - 1) y^2 + r_1^2 x^2 + } \\\\\\ + &= { + (x^2 + y^2) \left ( -x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2} \right ) + \over + (r_1^2 - 1) (x^2 + y^2) + } \\\\\\ + &= \left(\sqrt{(r_1^2 - 1) y ^2 + r_1^2 x^2} - x\right) / (r_1^2 - 1) +\end{align} + +The transformation above (multiplying $-x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}$ to enumerator and +denomenator) is always valid because $r_1 > 1$ and it's the unique solution due to Corollary 1. +$\square$ + +**Lemma 4.** If $r_1 < 1$, then + +1. there's no solution to $x_t$ if $(r_1^2 - 1) y^2 + r_1^2 x^2 < 0$ +2. otherwise, the solutions are + $x_t = \left(\sqrt{(r_1^2 - 1) y ^2 + r_1^2 x^2} - x\right) / (r_1^2 - 1)$, + or + $x_t = \left(-\sqrt{(r_1^2 - 1) y ^2 + r_1^2 x^2} - x\right) / (r_1^2 - 1)$. + +(Note that solution $x_t$ still has to be nonnegative to be valid; also note that +$x_t > 0 \Leftrightarrow x > 0$ if the solution exists.) + +*Proof.* Case 1 follows naturally from Lemma 3. and Corollary 1. + +<img src="conical/lemma4.svg"/> + +For case 2, we notice that $||C_f P_1||$ could be + +1. either $||C_f H|| + ||H P_1||$ or $||C_f H|| - ||H P_1||$ if $x \geq 0$, +2. either $-||C_f H|| + ||H P_1||$ or $-||C_f H|| - ||H P_1||$ if $x < 0$. + +By analysis similar to Lemma 3., the solution to $x_t$ does not depend on the sign of $x$ and +they are either $\frac{x^2 + y^2}{x + \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}}$ +or $\frac{x^2 + y^2}{x - \sqrt{(r_1^2 - 1) y^2 + r_1^2 x^2}}$. + +As $r_1 \neq 1$, we can apply the similar transformation in Corollary 3. to get the two +formula in the lemma. +$\square$ diff --git a/site/dev/design/conical/lemma1.ggb b/site/dev/design/conical/lemma1.ggb Binary files differnew file mode 100644 index 0000000000..1e80fe0b2e --- /dev/null +++ b/site/dev/design/conical/lemma1.ggb diff --git 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