Curve implicitization using moving lines
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Curve implicitization using moving linesThomas W. Sederberg, Takafumi Saito, Dongxu Qi, Krzysztof S. Klimaszewski (1993)1Presented by: Mira ShalahIntroduction, motivation and definitionsPencils of linesMoving lines2Curve implicitization with two moving linesImplicitization3Implicitization of 2-D curves leads to many practical algorithms.
For example, a very fast algorithm for computing the intersection of 2-D curves of low degree is based on implicitization.
Implicitization reduces the problem of curve intersection to one of finding the roots of a single polynomial.
Implicitization- why ?4The standard method5Contributions6Some Definitions78Dualitymoving pointmoving line9101011Pencil of lines
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Intersection of two pencils1314
15Pencils on quadratic curves1617
Moving lines: Bernstein form1819Intersection of two moving linesBase Points and Axial moving lines20Curve representation with two moving lines2122
Axial moving line on a double point.23In general:24Cubic curves25
2627Quartic curves
Any quartic rational Bezier curve can be expressed either with two quadratic28Cont.29
30The general case3132Cont.3334
The zeroing process3536Implicitization373738The conventional methodThe new methodfaster39?