Geodesic curve computations on surfaces

doi:10.1016/S0167-8396(03)00023-2

Computer Aided Geometric Design

Volume 20, Issue 2, May 2003, Pages 119-133

https://doi.org/10.1016/S0167-8396(03)00023-2 Get rights and content

Abstract

A study on geodesic curves computed directly on NURBS surfaces and discrete geodesics computed on the equivalent tessellated surfaces has been presented. A new approach has been presented for the computation of discrete geodesics on tessellated surfaces. An available approach has been extended for the computation of geodesics on NURBS surfaces. The new approach for the computation of discrete geodesics takes into account the tessellation normal and compares better with the geodesic curves computed directly on the NURBS surface. Many geodesics computed using these approaches on both developable and non-developable surfaces and their equivalent tessellated surfaces have been presented.

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Visualizing geodesics

There are more references available in the full text version of this article.

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The improved CH algorithm used triangular meshes to model surfaces and computed the geodesic between two points, reducing computation cost greatly compared with the original one. Kumar et al. [35] suggested two approaches for numerical computation of geodesics. A geodesic was calculated on surface tessellation from a given start point and a specified geodesic direction vector.
In this paper, a quadrilateral grid generation algorithm based on surface fitting is proposed for complex free-form surfaces. The ray-surface intersection and plane-surfce intersection methods are proposed to fit complex free-form surfaces into one free-form surface. Based on the fitted surface, quadrilateral grids were generated by using the geodesics or the Chebyshev algorithm. The grids were then smoothed by a physics-based algorithm. The grid quality was evaluated by fluency, shape quality and rod length indices. Compared with the conventional method in which grids are separately generated and manually modified, the fluency index and the standard deviation of the rod length index of the grids generated by the proposed methods reduce by up to 87% and 94% respectively. The results show that the grids generated by the proposed methods are more fluent and uniform compared with those by the conventional method.
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Therefore, an iterative forward algorithm inspired partially by the work performed by Surazhsky et al. [28], Kumar et al. [15], and Kimmel and Sethian [29] on calculating geodesics for triangular meshes was created. Kumar et al. [15] describe two methods, method2a and method2b, which are most similar to the approach developed in this work. The approach created to produce the results seen later in this study is a simple geodesic propagation algorithm expressed in Algorithm 1.
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Discrete Lagrangian algorithm for finding geodesics on triangular meshes
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The present paper introduces an approximation method for finding open geodesics on triangular surfaces. The algorithm is specifically designed to be able to solve real world problems where geodesic paths are needed. We use the model of geodesic curvature flow for open curves in the Lagrangian formulation. The model is enriched with a tangential term in order to have a control over the quality of the discretization grid during the computation. The governing equation of the flow is solved by a numerical method based on a semi-implicit time discretization and a finite difference space discretization. The paper presents the numerical scheme and various implementation details as well as numerous experiments to demonstrate the performance of the method and to provide comparison with several other well known methods. We also present a Grasshopper component for Rhinoceros for finding optimal paths on surface meshes that we developed and that includes our algorithm.
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Citation Excerpt :
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Computing geodesic paths and distances is a common operation in computer graphics and computer-aided geometric design. The existing discrete geodesic algorithms are mainly designed to solve the boundary value problem, i.e., to find the shortest path between two given points. In this paper, we focus on the initial value problem, i.e., finding a uniquely determined geodesic path from a given point in any direction. Since the shortest paths do not provide the unique solution on triangle meshes, we solve the initial value problem in an indirect manner: given a fixed point and an initial tangent direction on a triangle mesh $M$ , we first compute a geodesic curve $\hat{γ}$ on a piecewise smooth surface $\hat{M}$ , which well approximates the input mesh $M$ and can be constructed at little cost. Then, we solve a first-order ODE of the tangent vector using the fourth-order Runge–Kutta method, and parallel transport it along $\hat{γ}$ . When the geodesic curve reaches the boundary of the current patch, its tangent can be directly transported to the neighboring patch, thanks to the $G^{1}$ -continuity along the common boundary of two adjacent patches. Finally, once the geodesic curve $\hat{γ}$ is available, we project it onto the underlying mesh $M$ , producing the discrete geodesic path $γ$ , which is guaranteed to be unique on $M$ . It is worth noting that our method is different from the conventional methods of directly solving the geodesic equation (i.e., a second-order ODE of the position) on piecewise smooth surfaces, which are difficult to implement due to the complicated representation of the geodesic equation involving Christoffel symbols. The proposed method, based on the first-order ODE of the tangent vector, is intuitive and easy for implementation. Our method is particularly useful for computing geodesic paths on low-resolution meshes which may have large and/or skinny triangles, since the conventional straightest geodesic paths are usually far from the ground truth.

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Geodesic curve computations on surfaces

Abstract

Computer-Aided Design

Computer-Aided Design

Differential Geometry of Curves and Surfaces

Visualizing geodesics