Computer Graphics - Revision history

Adamcolton at 14:02, 28 August 2019

2019-08-28T14:02:03Z

2019-08-09T16:09:46Z

Polynomial Curve Construction (95)

← Older revision		Revision as of 16:09, 9 August 2019
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	Point-Normal interpolation.		Point-Normal interpolation.

	Least square approximations. We don't need the curve to pass through each point, just pass close.		Least square approximations. We don't need the curve to pass through each point, just pass close. Further, by relaxing the error requirements, we can clean up the control polygon.

	== B-spline ==		== B-spline ==

2019-08-08T16:37:51Z

Polynomial Curve Construction (95)

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 Given an ordered set of points, find a curve that passes through them. We define this as
    p(t<sub>i</sub>) = p<sub>i</sub>
 == B-spline ==

2019-08-08T16:17:28Z

Polynomial Curve Construction (95)

← Older revision		Revision as of 16:17, 8 August 2019
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	==== Polynomial Curve Construction (95) ====		==== Polynomial Curve Construction (95) ====
	~~I've skimmed a lot~~ of ~~this~~, ~~but I'll need to go back to the beginning~~.		Given an ordered set of points, find a curve that passes through them. We define this as
			p(t<sub>i</sub>) = p<sub>i</sub>

	== B-spline ==		== B-spline ==

2019-07-24T02:20:34Z

Bezier Curve Topics (81)

← Older revision		Revision as of 02:20, 24 July 2019
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	Degree reduction is possible with a linear equation solver.		Degree reduction is possible with a linear equation solver.

			==== Polynomial Curve Construction (95) ====
			I've skimmed a lot of this, but I'll need to go back to the beginning.

	== B-spline ==		== B-spline ==

2019-07-24T00:19:06Z

Bezier Curve Topics (81)

← Older revision		Revision as of 00:19, 24 July 2019
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	A Bezier curve has no more intersections with a plane than it's control polygons.		A Bezier curve has no more intersections with a plane than it's control polygons.

			Degree reduction is possible with a linear equation solver.

	== B-spline ==		== B-spline ==

2019-07-24T00:12:09Z

Bezier Curve Topics (81)

← Older revision		Revision as of 00:12, 24 July 2019
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	Arbitrary degree elevation		Arbitrary degree elevation
	b<sub>i</sub><sup>+r</sup> = ∑ b<sub>j</sub> * bino(n,j) * bino(r, i-j) / bino(n+r, i)		b<sub>i</sub><sup>+r</sup> = ∑ b<sub>j</sub> * bino(n,j) * bino(r, i-j) / bino(n+r, i)

			A Bezier curve has no more intersections with a plane than it's control polygons.

	== B-spline ==		== B-spline ==

2019-07-23T22:46:01Z

← Older revision		Revision as of 22:46, 23 July 2019
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	A Bezier curve can also be expressed as a matrix.		A Bezier curve can also be expressed as a matrix.

	=== Bezier Curve Topics (81) ===		==== Bezier Curve Topics (81) ====
	A curve can be elevated to produce the same curve but with one more control point		A curve can be elevated to produce the same curve but with one more control point
	b<sub>i</sub><sup>+1</sup> = (i/(n+1)) * b<sub>i-1</sub> + ( 1 - (i/(n+1)) )*b<sub>i</sub>		b<sub>i</sub><sup>+1</sup> = (i/(n+1)) * b<sub>i-1</sub> + ( 1 - (i/(n+1)) )*b<sub>i</sub>

2019-07-23T22:45:39Z

The Bernstein Form (57)

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    c = t @ b6
    (3/(b-a))*(b3-b2) = (3/(c-b))*(b4-b3)
 == B-spline ==

2019-07-23T02:31:38Z

The Bernstein Form (57)

← Older revision		Revision as of 02:31, 23 July 2019
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	Line intersections can be found with repeated subdivision. If a line does not intersect the control polygon, it does not intersect the curve. If it does intersect the polygon, subdivide and recheck.		Line intersections can be found with repeated subdivision. If a line does not intersect the control polygon, it does not intersect the curve. If it does intersect the polygon, subdivide and recheck.

	Composite Bezier curves can be formed by piecing many curves together. Often these are cubic segments.		Composite Bezier curves can be formed by piecing many curves together. Often these are cubic segments. Given points b0 to b6 forming two composite curves (both containing b3), for the composite to be C<sup>1</sup>, and
			a = t @ b0
			b = t @ b3
			c = t @ b6
			(3/(b-a))(b3-b2) = (3/(c-b))(b4-b3)

	== B-spline ==		== B-spline ==