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Typo fix (#309)

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`k` is confusingly called `c` several times throughout this section
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Seph De Busser
2021-06-06 01:19:52 +02:00
committed by GitHub
parent a4511d217d
commit 6269b502ed
1 changed files with 4 additions and 4 deletions
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8
docs/chapters/circles_cubic/content.en-GB.md
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@@ -17,13 +17,13 @@ So let's again formally describe this:
\[
\begin{aligned}
P_1 &= (1, 0) \\
P_2 &= (1, c) \\
P_2 &= (1, k) \\
P_3 &= P_4 + k \cdot (sin(θ), -cos(θ)) \\
P_4 &= (cos(θ), sin(θ))
\end{aligned}
\]
Only P<sub>3</sub> isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P<sub>4</sub>, P<sub>3</sub>) is a right angled triangle, with the distance between the origin and P<sub>4</sub> being 1 (because we're working with a unit circle), and the distance between P<sub>4</sub> and P<sub>3</sub> being _c , so that we can represent P<sub>3</sub> as "The point P<sub>4</sub> plus the vector from the origin to P<sub>4</sub> but then rotated a quarter circle, counter-clockwise, and scaled by _c_".
Only P<sub>3</sub> isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P<sub>4</sub>, P<sub>3</sub>) is a right angled triangle, with the distance between the origin and P<sub>4</sub> being 1 (because we're working with a unit circle), and the distance between P<sub>4</sub> and P<sub>3</sub> being _k_, so that we can represent P<sub>3</sub> as "The point P<sub>4</sub> plus the vector from the origin to P<sub>4</sub> but then rotated a quarter circle, counter-clockwise, and scaled by _k_".
With that, we can determine the _y_-coordinates for A, B, e<sub>1</sub>, and e<sub>2</sub>, after which we have all the information we need to determine what the value of _k_ is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P<sub>2</sub> and P<sub>3</sub>, e<sub>1</sub> is between A and "midway between P<sub>1</sub> and P<sub>2</sub>" (which is "half height" P<sub>2</sub>), and so forth:
@@ -48,7 +48,7 @@ Solving for _k_ is fairly straight forward, but it's a fair few steps, and if yo
\[
\begin{aligned}
\frac{3c + 4sin(θ)) - 3k \cdot cos(θ)}{8} &= sin(\frac{θ}{2}) \\
\frac{3k + 4sin(θ)) - 3k \cdot cos(θ)}{8} &= sin(\frac{θ}{2}) \\
3k + 4sin(θ)) - 3k \cdot cos(θ) &= 8sin\left(\frac{θ}{2}\right) \\
3k - 3k \cdot cos(θ) &= 8sin\left(\frac{θ}{2}\right) - 4sin(θ) \\
3k (1 - cos(θ)) &= 4 \left ( 2sin\left(\frac{θ}{2} \right) - sin(θ) \right ) \\
@@ -306,4 +306,4 @@ However, always remember that if you're writing for humans, you can typically us
**If you're writing code for graphics design by humans, use circular arcs for circular arcs**
And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best _k_ values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).
And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best _k_ values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).