Curve Offset

1. Curve Offset Planar Curve Offset Based on Circle Approximation

2. Curve Offset • Planar Curve Offset Based on Circle Approximation – Lee, Kim, Elber. • Concept • Circle Approximation • Offset Approximation • Eliminating Self Intersecting Loops • Results • Comparison of offset approximation methods

3. Concept Given a planar regular parametric curve with normal the offset curve is . The offset curve is generally not rational, and cannot be described as a rational B-spline.

4. Concept The article suggests an approximation , calculated as the envelope of a convolution of the original curve and an approximated circle . This is achieved by adding to each point on the curve a specific point on the approximated circle of radius : is a reparameterization that keeps and at the same direction. This entails that is normal to , and therefore an approximation of .

5. Circle Approximation A quadratic Bezier curve is given by: + Assuming , because of symmetry:

6. Circle Approximation An alternative measure for the error, instead of : Requiring for extremal error, there are five solutions:

7. Method 1: Tangent to the circle at both ends If each quadratic Bezier curve is tangent to the circle at its endpoints, the whole piecewise curve is of continuity. This means that the middle control point of should be , and therefore . The resulting error has extremal values at the endpoints, minimal with value 0. The maximum error is in the middle at :

8. Method 2: Uniform scaling of Method 1 The error of is always positive, and it is outside of the unit circle. It is possible to minimize the maximal error by uniform scaling of the curve by some constant : The choice of affects the error function: The extremaare at the same values of , so by setting the minimal is achieved.

9. Method 2: Uniform scaling of Method 1 the value of is determined: The size of the error is slightly less than half of : Since the scaling factor depends on , the piecewise curve will only remain continuous (and preserve continuity) if all segments share the same .

10. Method 3: Interpolating three circle points If the continuity restriction is lifted, The middle point can be positioned at the mid-point of the arc: The error at is now , and the maximum error is

11. Method 4: Interpolation with equi-oscillating Error Requiring the same magnitude for maximal and minimal error: This determines the value of and the magnitude: =

12. Method 5: Uniform scaling of Method 3 The error of is always positive, and it is inside the unit circle. It is possible to minimize the maximal error by uniform scaling of the curve by some constant : The choice of affects the error function: Again, the extrema are at the same values of , so by setting the minimal is achieved.

13. Method 5: Uniform scaling of Method 3 the value of is determined: The size of the error is slightly more than half of : Since the scaling factor depends on , the piecewise curve will only remain continuous if all segments share the same .

14. Circle Approximation

15. Offset Approximation The purpose of defining the approximated arc segments was providing a quadratic equation to be re-parameterized in order to add it to the original curve . Different segments are relevant at different values of , so the adding is done separately for each continuous part of that has a single corresponding approximated arc.

16. Hodograph Definitions: is a planar regular parametric curve. • The hodograph curve is the locus of . • The tangential angular map of is

17. Hodograph Lemma 1: Let be the hodograph of . If the tangential angular map of is one-to-one, any ray from the origin intersects with at no more than one point. Proof: If intersects with at two different points and , , then and have the same ratio as and , implying =

18. Hodograph Lemma 2: • is the hodograph of • is the hodograph of If and are intersection points of a ray starting from the origin, Then and have the same tangent direction at and .

19. Hodograph Proof: The direction of is the direction of vetors and , therefore and have the same direction.

20. Approximated Offset Curve • is one-to-one. • is the ray from origin through By the first lemma, • intersects with at the point , This defines a mapping from to . By the second lemma, and are in the same direction, and therefore the curve is indeed the well-defined convolution curve needed.

21. Approximated Offset Curve is quadratic, and it’s hodograph curve is linear: By demanding the same direction for and :

22. Approximated Offset Curve + If is a polynomial of degree : is a rational polynomial of degree is a rational polynomial of degree is a rational curve of degree If is a rational polynomial of degree : is a rational polynomial of degree is a rational polynomial of degree is a rational curve of degree

23. Subdivision of Until now it was assumed that , and was an approximation of an arc from angle 0 to . Several can be connected to approximate a whole circle. is subdivided into , where each part satisfies

24. Subdivision of Hodograph of : Hodograph of : Offset curve:

25. Subdivision of If the circle was approximated in methods 3, 4 or 5, the hodograph is not continuous. This can be solved by adding a zero-radius arcs between the intervals, which become arcs in the hodograph. The discontinuity of tangent angles at the Endpoints of segmentsof Q(s):

26. Subdivision of The error of the approximation is determined by the choice of circle approximation method and by the choice of .

27. Eliminating self intersecting loops • Original curve • Sampled points • Offset of sampled points • Offset only of segments with curvature • Intersections by Plane Sweep • Valid offset curve

28. results

29. results

30. results

31. Comparisons Control-polygon based methods: • Cobb – translation of control-points in normal direction. • Tiller and Hanson – translation of control segments • Coquillart– translation of control points using closest normal to curve • Elber and Cohen – error minimization of new control points

32. Comparisons Interpolation methods: • Hoschek– least squares of errors, parallel endpoints.

33. Comparisons Quadratic Polynomial

34. Comparisons Cubic Polynomial

35. Comparisons Cubic Polynomial

36. Comparisons Quadratic Polynomial

37. Comparisons Results: • Least Square Error performs well on general curves. • Tiller and Hanson is very good for quadratic curves. • The best-performing geometrical method for general curve is circle approximation.