Curve implicitization using moving lines

1. Curve implicitization using moving lines Thomas W. Sederberg, Takafumi Saito, Dongxu Qi, Krzysztof S. Klimaszewski (1993) Presented by: Mira Shalah

2. Introduction, motivation and definitions Pencils of lines Moving lines Curve implicitization with two moving lines

3. Implicitization For any 2-D parametric curve , (,, are polynomials), there exists an implicit equation , s.t is also a polynomial, which defines exactly the same curve. Example: a circle can be defined by the parametric equation , or by the implicit equation . The process of finding the implicit equation given the parametric equations is known as implicitization.

4. Implicitization- why ? Implicitization 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.

5. The standard method The standard method for implicitizing a 2-D curve is to use Bezout’s resultant. For a degree rational curve, Bezout’s resultant is the determinant of an matrix whose elements are linear in .

6. Contributions It is a classical result that two corresponding pencils of lines intersect in a conic section. This paper presents an extension to this idea to higher degree families of lines. We will see that any planar rational curve can be expressed as the intersection of two families of lines. This extension leads to a more efficient implicitization algorithm for curves, in which, for example, the implicit equation of a degree 4 rational curve can generally be expressed as the determinant of a matrix.

7. Some Definitions Homogeneous coordinate Cartesian coordinate The equation of a line in homogenous form is Cross product: Dot product:

8. Duality • A point lies on the line iff . • The line containing two points and • The point at which two lines and intersect:

9. A homogeneous point whose coordinates are functions of a variable Which amounts to the rational curve Denotes the family of lines moving point moving line

10. A moving point follows a moving line if That is, if point lies on the line for all values of . Two moving points and follow a moving line if Two moving lines and intersect at a moving point if

11. Pencil of lines The parameter line is a degree one polynomial Bezier curve which follows passes through the parameter line at the point corresponding to . Thus, every pencil of lines follows a degree 1 polynomial curve.

12. Intersection of two pencils To each value of corresponds exactly one line from each pencil and those two lines intersect in a point. The locus of the points created for is a conic section.

13. This conic section can be expressed as a rational Bezier curve .

14. Which expresses as a quadratic rational Bezier curve whose control points are

15. The two pencils provide an intermediate way to represent of a conic section: In matrix form, So the implicit equation of the intersection locus is:

16. Pencils on quadratic curves An arbitrary rational quadratic Bezier curve: Can be represented as the intersection of two pencils. Consider a moving line which goes through a certain fixed point on the curve and follows the moving point .

17. is equivalent to a linear moving line (pencil). Thus, two pencils can be found by choosing arbitrary two parameter values and and calculating

18. Moving lines: Bernstein form If the polynomials and are polynomials of degree , one way to define the moving line is to use control lines where

19. Intersection of two moving lines Two moving lines and of degree and respectively, intersect at a moving point Whose degree is generally .

20. Base Points and Axial moving lines Any parameter value for which or is called a base point. Any curve or moving line which has a base point can be replaced by an equivalent curve or moving point of degree one less. When the line has an axis (a single point at which it rotates) then the line is referred to as an axial moving line. A degree axial moving line can be expressed as:

21. Curve representation with two moving lines Consider an axial moving line that follows a degree Bezier curve : If the axis lies on the curve , the degree of the moving line can be reduced to because then the moving line has a base point. In general, if the axis is on a point of multiplicity , then the degree of the axial moving line which follows the degree curve is .

22. Axial moving line on a double point. This cubic Bezier curve has a double point at . Thus, the axial moving line with the axis is a pencil:

23. For example, for cubic Beziers, Where () are the control points of the Bezier curve.

24. In general: Where The determinant of this matrix is equivalent to Bezout’s resultant.

25. Cubic curves For any rational cubic Bezier curve, we can find a pencil and a quadratic moving line which intersect at the moving points. Let’s calculate them for our example: We first make axial moving lines which follow the moving point:

26. This moving line follows for any , so the following equation is obtained: Now the bottom moving line has a base point at and it is reducible to a linear moving line

27. Quartic curves Any quartic rational Bezier curve can be expressed either with two quadratic moving lines, or with one linear and one cubic moving line. Calculate four cubic moving lines which follow the curve by:

28. Cont. From which we can obtain two quadratic moving lines: That follow the curve, and hence which intersect at the curve. If the moving line is axial, then the axis is a double point.

29. If there is a triple point on the curve, there is no pair of quadratic curves that can represent the curve. For example, This curve has a double point at We can obtain four cubic moving lines that follow the curve. From which we can obtain the two moving lines:

30. These two quadratic moving lines (call them and ) are linearly independent, but their intersection does not express the curve because all control point obtained from are In fact, each of them has a base point and both are identical to the same pencil. The pencil can be obtained by eliminating the bottom left element:

31. The general case The method for cubic curve can be extended to higher degree curves. For a degree curve, there exists an parameter family of moving lines whose degree is . A basis for that family of moving lines is given by Where

32. By calculating linear combinations of these rows of the matrix, we can always zero out all but two elements in the first column:

33. Cont. denotes the element after the -thcalculation. Now, we have moving lines whose degree is : In this matrix, it turns out that we can again zero out all but two elements of the first column, since, all the lines contain the point !

34. Magic! Why is that? By evaluating our set of equations at we get: This zeroing process can be repeated until one or two rows remain!

35. The zeroing process If the degree is , we can repeat the element zeroing process times and obtain two degree moving lines.

36. If the degree is , we can repeat the zero out process times. This time, only one element can be eliminated in the last step, and thus, the bottom two rows express degree and degree moving lines

37. Implicitization We can implicitize a rational Bezier curve using a pair of moving lines. A point lies on the curve if and only if it lies on both moving lines. This is equivalent to saying that if lies on the curve, then there exists a value of which satisfies both equations, which means that the resultant of the two equations is zero.

38. faster determinant determinant The conventional method The new method

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