Author Contributions
Conceptualization, T.S. and G.S.; Methodology, H.L., P.K.V. and T.S.; Software, H.L., P.K.V. and T.S.; Validation, H.L., P.K.V., T.S. and G.S.; Formal analysis, P.K.V.; Investigation, H.L., P.K.V., T.S. and G.S.; Resources, G.S.; Data curation, H.L., P.K.V., and T.S.; Writing—original draft preparation, H.L., P.K.V., T.S. and G.S.; Writing—review and editing, G.S.; Visualization, H.L., P.K.V. and T.S.; Supervision, G.S.; Project administration, G.S.; Funding acquisition, G.S. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Special cases encountered during point projection: (a) Exact projection from points on a straight line to a bell-shaped curve. Discontinuity occurs at the circled central point due to non-uniqueness of point projection. (b) Projection from points on a straight line to a cone-shaped curve. The dashed line segment has no footpoint on the curve.
Figure 1.
Special cases encountered during point projection: (a) Exact projection from points on a straight line to a bell-shaped curve. Discontinuity occurs at the circled central point due to non-uniqueness of point projection. (b) Projection from points on a straight line to a cone-shaped curve. The dashed line segment has no footpoint on the curve.
Figure 2.
Behavioral analysis in the presence of complex free form embedded surface. The spatial point may only influence a local region of the surface with the highlighted control points, which can be identified by point projection.
Figure 2.
Behavioral analysis in the presence of complex free form embedded surface. The spatial point may only influence a local region of the surface with the highlighted control points, which can be identified by point projection.
Figure 3.
Implicitization of a quartic Bézier curve. Level set can be used as a measure of distance.
Figure 3.
Implicitization of a quartic Bézier curve. Level set can be used as a measure of distance.
Figure 4.
A convex region is used to trim the implicitized curve constructed from a parametric curve .
Figure 4.
A convex region is used to trim the implicitized curve constructed from a parametric curve .
Figure 5.
The control points of a cubic Bézier curve form a convex hull consisting of four hyper-planes and with inner normals , and , respectively. Boolean intersection of the four hyper-planes using R-functions yields a trimming region .
Figure 5.
The control points of a cubic Bézier curve form a convex hull consisting of four hyper-planes and with inner normals , and , respectively. Boolean intersection of the four hyper-planes using R-functions yields a trimming region .
Figure 6.
Algebraic level sets of a symmetric cubic spline. continuity is present at and . The generated algebraic level sets retain the symmetry while ensuring the smoothness of the field.
Figure 6.
Algebraic level sets of a symmetric cubic spline. continuity is present at and . The generated algebraic level sets retain the symmetry while ensuring the smoothness of the field.
Figure 7.
Algebraic level sets from a symmetric quadratic NURBS surface. (a) The valley of the surface contains only a continuity across the plane of symmetry. The level sets are plotted over three principal planes slicing the surface: (b) x-y plane; (c) y-z plane; and (d) x-z plane.
Figure 7.
Algebraic level sets from a symmetric quadratic NURBS surface. (a) The valley of the surface contains only a continuity across the plane of symmetry. The level sets are plotted over three principal planes slicing the surface: (b) x-y plane; (c) y-z plane; and (d) x-z plane.
Figure 8.
First-order algebraic point projection fails to reach footpoints on the curve for test points not close to quadratic Bézier curve.
Figure 8.
First-order algebraic point projection fails to reach footpoints on the curve for test points not close to quadratic Bézier curve.
Figure 9.
Point projection for cubic Bézier curve in two-dimensional physical space using the developed algebraic method as well as Newton–Raphson iterations for: (a) test distance ; and (b) test distance . From left to right, Newton–Raphson method, first-order algebraic point projection, and second-order algebraic point projection are shown. The inset image shows that first-order point projection fails to converge onto footpoints on the curve when distances are larger.
Figure 9.
Point projection for cubic Bézier curve in two-dimensional physical space using the developed algebraic method as well as Newton–Raphson iterations for: (a) test distance ; and (b) test distance . From left to right, Newton–Raphson method, first-order algebraic point projection, and second-order algebraic point projection are shown. The inset image shows that first-order point projection fails to converge onto footpoints on the curve when distances are larger.
Figure 10.
Recursive first-order algebraic point projection at test distance .
Figure 10.
Recursive first-order algebraic point projection at test distance .
Figure 11.
Second-order algebraic point projection with test points at a far distance: (a) using non-corrected normal vector; and (b) using corrected normal vector.
Figure 11.
Second-order algebraic point projection with test points at a far distance: (a) using non-corrected normal vector; and (b) using corrected normal vector.
Figure 12.
Illustration of the second projection onto an adjacent Bézier curve segment if the first projection yields an out-of-span solution.
Figure 12.
Illustration of the second projection onto an adjacent Bézier curve segment if the first projection yields an out-of-span solution.
Figure 13.
Point projection in 3D physical space using the proposed algebraic method and Newton–Raphson iterations.
Figure 13.
Point projection in 3D physical space using the proposed algebraic method and Newton–Raphson iterations.
Figure 14.
Illustration of the second projection onto an adjacent Bézier surface segment if the first projection yields an out-of-span solution.
Figure 14.
Illustration of the second projection onto an adjacent Bézier surface segment if the first projection yields an out-of-span solution.
Figure 15.
Flowchart for execution of algebraic point projection.
Figure 15.
Flowchart for execution of algebraic point projection.
Figure 16.
Parameter values of footpoints obtained using: (a) Newton–Raphson method; (b) first-order algebraic point projection; and (c) second-order algebraic point projection. Parameter range of NURBS curve is .
Figure 16.
Parameter values of footpoints obtained using: (a) Newton–Raphson method; (b) first-order algebraic point projection; and (c) second-order algebraic point projection. Parameter range of NURBS curve is .
Figure 17.
Relative error vs. distance of test points , where and are parameter values of footpoints obtained using the Newton–Raphson method and the algebraic point projection respectively.
Figure 17.
Relative error vs. distance of test points , where and are parameter values of footpoints obtained using the Newton–Raphson method and the algebraic point projection respectively.
Figure 18.
Illustration of the robustness of the 2D algebraic point projection for the NURBS curve. (a) Trace of points that were projected onto target curve. (b) Solution parameter of footpoints on target curve vs. parameter of trace curve for the two methods. Parameter discontinuity in Newton–Raphson solution occurs due to non-uniqueness of the footpoint near the local minimum at
Figure 18.
Illustration of the robustness of the 2D algebraic point projection for the NURBS curve. (a) Trace of points that were projected onto target curve. (b) Solution parameter of footpoints on target curve vs. parameter of trace curve for the two methods. Parameter discontinuity in Newton–Raphson solution occurs due to non-uniqueness of the footpoint near the local minimum at
Figure 19.
Illustration of the robustness of the 3D algebraic point projection algorithm involving discontinuous footpoints. (a) Trace of points that were projected onto bowl-shaped target surface using the proposed algebraic method and the Netwon-Raphson method. (b) Parameters of footpoints on the target obtained by both the methods. Discontinuity occurs due to non-unique footpoints for test points near the bottom of the surface.
Figure 19.
Illustration of the robustness of the 3D algebraic point projection algorithm involving discontinuous footpoints. (a) Trace of points that were projected onto bowl-shaped target surface using the proposed algebraic method and the Netwon-Raphson method. (b) Parameters of footpoints on the target obtained by both the methods. Discontinuity occurs due to non-unique footpoints for test points near the bottom of the surface.
Figure 20.
Illustration of the robustness of the 3D algebraic point projection algorithm involving test points whose mathematical footpoints do not exist. (a) Trace of points which are projected onto mountain-shaped target surface using both methods. (b) Parameters of footpoints on target. The solution does not exist near the four mountain ridges of continuity as shown in the four corner regions of (b).
Figure 20.
Illustration of the robustness of the 3D algebraic point projection algorithm involving test points whose mathematical footpoints do not exist. (a) Trace of points which are projected onto mountain-shaped target surface using both methods. (b) Parameters of footpoints on target. The solution does not exist near the four mountain ridges of continuity as shown in the four corner regions of (b).
Table 1.
Comparison of methods used for point projection in literature
Table 1.
Comparison of methods used for point projection in literature
Description | Reference | Algebraic | Initial Guess Dependent? | Efficiency | Accuracy | Smoothness |
---|
Subdivision method | [22] | Yes | No | High | Medium | No |
Subdivision method + Newton–Raphson method | [12,14,15] | No | Yes | High | High | No |
Geometric iteration method | [16,17,18,19,20,21] | No | Yes | High | High | No |
Proposed method | | Yes | No | High | Medium | Yes |
Table 2.
Time complexity of each step in Algorithm 1 for computing the algebraic level sets for a Bézier segment. Time complexities are listed for Bézier curves of degree p and Bézier surfaces of degree .
Table 2.
Time complexity of each step in Algorithm 1 for computing the algebraic level sets for a Bézier segment. Time complexities are listed for Bézier curves of degree p and Bézier surfaces of degree .
Step | Time Complexity |
---|
Curve | Surface |
---|
Convex hull construction | | |
Distance field of convex hull | | |
Computing Dixon resultant | | |
Normalization of resultant | | |
Trimming operation | | |
Table 3.
Time complexity of each step in Algorithm 2 for algebraic point projection. Time complexities are listed for NURBS curves of degree p and NURBS surfaces of degree .
Table 3.
Time complexity of each step in Algorithm 2 for algebraic point projection. Time complexities are listed for NURBS curves of degree p and NURBS surfaces of degree .
Step | Time Complexity |
---|
Curve | Surface |
---|
Computing algebraic level set | | |
Determining the closest Bézier segment | | |
Projection in physical space | | |
Point inversion to parametric space | | |
Scaling and offset | | |
Table 4.
The results of point projection for NURBS surfaces. The tolerance in Newton–Raphson iterations was chosen as . Note that the time of finding an initial point is excluded in the time per iteration.
Table 4.
The results of point projection for NURBS surfaces. The tolerance in Newton–Raphson iterations was chosen as . Note that the time of finding an initial point is excluded in the time per iteration.
Example Surface | Newton–Raphson Iterations | 2nd order APP |
---|
Time per Point (s) | Average Number of Iterations | Time per Iteration (s) | Time per Point (s) | Time per Point for Algorithm 1 (s) |
---|
#1 (Figure 19) | 211.91 | 5.00 | 36.03 | 55.28 | 14.84 |
#2 (Figure 20) | 769.73 | 10.85 | 50.47 | 107.88 | 16.92 |