Michael Kallay

Monotone linear rational spline interpolation

Abstract A method for interpolating a monotone data sequence (values and derivatives) with a C1 monotone rational B-spline of degree 1 is presented, along with some test results. This method is tailor-made for reparameterizing B-spline curves in Geometric Modeling.

Computing the Moment of Inertia of a Solid Defined by a Triangle Mesh

This note presents a simple algorithm for computing the mass properties of a solid whose boundary is defined by a mesh of triangles. Its main contribution is a simplification of the formula for computing moment of inertia. Source code is available online at the website listed at the end of this paper.

A geometric Newton---Raphson strategy

Abstract In the standard Newton–Raphson algorithm for solving nonlinear equations, a new guess is computed by solving a linear approximation of the problem at the current guess. A similar, very effective strategy is proposed here for solving geometric problems (e.g., finding intersections) on general plane curves. To compute a new guess, solve an elementary geometric approximation of the problem at the current guess.

Object-oriented paradigm for NURBS curve and surface design

Abstract The prevailing paradigm in the design of curve and surface algorithms could be described as being control-points oriented. The limitations of this approach are explored, and an alternative paradigm is presented that combines object-oriented design with two well known approximation algorithms that are rarely mentioned in the cad literature. It provides a powerful and versatile tool for constructing NURBS entities.

Geometric algorithms on an ellipsoid earth model

This paper outlines some geometric algorithms that are used in Microsoft SQL Server for processing geospatial data in a way that is independent of any cartographic projection.

Defining edges on a round earth

On planar maps lines, polygons and polygonal paths are clearly defined by their endpoints and vertices, but on a round earth the definition of the connecting edges is far from obvious. This paper presents a definition, examines its properties, and proposes it as an industry standard. Algorithms based on this definition are outlined.

Computing Tight Bounds for a Bézier Curve

Abstract This note shows how to compute tight bounds (within ar ithmetic error) on cubic Bézier curves. A C++ implementation is available online.

Defining circular arcs on a round earth

Circular arcs are commonly used for modeling geospatial data in planar map projections. This paper proposes a definition of circular arcs in geodetic coordinates on ellipsoid earth models.

Improving the Two-Pass Resampling Algorithm

Abstract The Catmull-Smith two-pass resampling algorithm simplifies the nontrivial reconstruction of a transformed two-dimensional image by decomposing the transformation into two one-dimensional passes. We present a theoretically motivated modification to this algorithm that provides improved image quality. For the case of projective transformations, this improvement results in a final algorithm that is more robust and accurate than the original while even affecting correctness in some cases.