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BendSketch: Modeling Freeform Surfaces Through 2D Sketching

CHANGJIAN LI, The University of Hong Kong and Microsoft Research Asia HAO PAN, YANG LIU, and XIN TONG, Microsoft Research Asia

ALLA SHEFFER, University of British Columbia

WENPING WANG, The University of Hong Kong

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Introduction

Sketch-based modeling provides a powerful paradigm for geometric modeling
While existing sketching tools, they are not well suited for designing free form surface with complex curvature patterns

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Surface Sketching Pipeline

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Stroke Types

Boundry
Bending stroke
Sharp feature
Ridge/Valley
Flat stroke
Contour

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Stroke Parsing

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Relation of Convex/Concave Properties

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Domain Triangulation

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Surface from curvature fieild

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Curvature Field Computation

Curvature direction field : regulate the principal curvature directions of a smooth surface as projected on a plane [Iarussi et al. 2015]

Curvature magnitude field : regulate the principal curvature values through minimizing the following energy

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Joint Optimization of Surface and Curvature

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Estimating Principal Curvatures

To estimate the principal curvature along a bending curve, we lift the planar curve to the current spatial surface
For each point in the lifted curve, the normal

section plane is spanned by the surface nor-

mal N at the point and the curve tangent T.

On which we project a neighborhood of the spatialcurve, and fit a circle to the projected image.

The neighborhood size to estimate curvature with should be chosen:

if the neighborhood is too big, the estimated curvature can be too conservative and inexible,

while a size too small causes instability and inaccuracy due to the locally noisy shape of the lifted stroke

For fast computation, we simply divide a lifted stroke into segments of approximate length D/30

and estimate the curvature for each segment

which is then assigned to stroke points belonging to the segment

To carry out the circle tting for curvature estimation,

we minimize the standard geometric tting error, plus a regularization of radius for robustness:

最後一張圖片(公式)的註解:

where {pi } are the points to be tted,

(c,r) dene the circle,

and µ = 2 ×10−5 is a very small weight.

This optimization is a nonlinear least square problem with only 3 variables and is solved by GaussNewton iterations.

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User Adjustment

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Boundary Conditions

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Result

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Result

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Result

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Result

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Comparison

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Comparison

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Comparison

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Appendix

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Appendix