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Fixes to moulding section (#198)

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* “original points B” -> “original point B”
* punctuation fix.
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Simon Cozens 2019-06-11 02:05:53 +01:00 committed by Pomax
parent 6a61608ca9
commit 3e8690e761
1 changed files with 2 additions and 2 deletions
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components/sections/moulding/content.en-GB.md
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@ -4,7 +4,7 @@ Armed with knowledge of the "ABC" relation, we can now update a curve interactiv
<Graphic title="Moulding a quadratic Bézier curve" setup={this.setupQuadratic} draw={this.drawMould} onClick={this.placeMouldPoint} onMouseDown={this.markQB} onMouseDrag={this.dragQB} onMouseUp={this.saveCurve}/>
**Click-dragging the curve itself** shows what we're using to compute the new coordinates: while dragging you will see the original points B and its corresponding <i>t</i>-value, the original point C for that <i>t</i>-value, as well as the new point B' based on the mouse cursor. Since we know the <i>t</i>-value for this configuration, we can compute the ABC ratio for this configuration, and we know that our new point A' should like at a distance:
**Click-dragging the curve itself** shows what we're using to compute the new coordinates: while dragging you will see the original point B and its corresponding <i>t</i>-value, the original point C for that <i>t</i>-value, as well as the new point B' based on the mouse cursor. Since we know the <i>t</i>-value for this configuration, we can compute the ABC ratio for this configuration, and we know that our new point A' should like at a distance:
\[
A' = B' - \frac{C - B'}{ratio} = B' + \frac{B' - C}{ratio}
@ -14,7 +14,7 @@ For quadratic curves, this means we're done, since the new point A' is equivalen
<Graphic title="Moulding a cubic Bézier curve" setup={this.setupCubic} draw={this.drawMould} onClick={this.placeMouldPoint} onMouseDown={this.markCB} onMouseDrag={this.dragCB} onMouseUp={this.saveCurve}/>
To help understand what's going on, the cubic graphic shows the full de Casteljau construction "hull" when repositioning point B. We compute A` in exactly the same way as before, but we also record the final strut line that forms B in the original curve. Given A', B', and the endpoints e1 and e2 of the strut line relative to B', we can now compute where the new control points should be. Remember that B' lies on line e1--e2 at a distance <i>t</i>, because that's how Bézier curves work. In the same manner, we know the distance A--e1 is only line-interval [0,t] of the full segment, and A--e2 is only line-interval [t,1], so constructing the new control points is fairly easy.
To help understand what's going on, the cubic graphic shows the full de Casteljau construction "hull" when repositioning point B. We compute A' in exactly the same way as before, but we also record the final strut line that forms B in the original curve. Given A', B', and the endpoints e1 and e2 of the strut line relative to B', we can now compute where the new control points should be. Remember that B' lies on line e1--e2 at a distance <i>t</i>, because that's how Bézier curves work. In the same manner, we know the distance A--e1 is only line-interval [0,t] of the full segment, and A--e2 is only line-interval [t,1], so constructing the new control points is fairly easy.
First, we construct the one-level-of-de-Casteljau-up points: