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Approximation of a Parametric Bicubic Surface

IP.com Disclosure Number: IPCOM000077992D
Original Publication Date: 1972-Oct-01
Included in the Prior Art Database: 2005-Feb-25
Document File: 3 page(s) / 76K

Publishing Venue

IBM

Related People

Dimsdale, B: AUTHOR

Abstract

The object of the flow diagram (page 1652) is to approximate a parametric bicubic surface by a nonparametric bicubic surface, with patches of the form z = f(x,y). Once such a surface has been defined in this manner, machine tool cutter paths may be computed to reproduce the surface by nulling.

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Approximation of a Parametric Bicubic Surface

The object of the flow diagram (page 1652) is to approximate a parametric bicubic surface by a nonparametric bicubic surface, with patches of the form z = f(x,y). Once such a surface has been defined in this manner, machine tool cutter paths may be computed to reproduce the surface by nulling.

Input 10 to the flow diagram is assumed to consist, at a minimum, of an ordered m x n array of points, together with certain option parameters. It is assumed that the surface implied by this array is a parametric bicubic patch surface (Coons patches). It is also assumed that the surface implied by the data is single valued with respect to some plane (call it the x-y plane or the horizontal plane) and that the plane is specified. The flow diagram can either do the spline fitting implied by these assumptions, or accept patch specifications 11 provided by the user.

An m x n array of points defines (m-1) x (n-1) patches. A defining point of the surface is called a "knot", and generally connects four patches, unless it is a boundary knot. Conversely, a knot is a boundary knot if and only if it connects fewer than four patches. An edge is a spline segment between two knots, and generally connects two patches. Otherwise it is a boundary edge. A patch is specified by the indices of one of its four knots, specifically the knot of smallest indices.

As pointed out by test 12, if it happens that the points when projected onto the x-y plane define a rectangular grid, there is nothing more to be done, as shown by path 13, since the object is to define an approximating surface over a rectangular grid.

If this is not the case 14, a rotation 15 of the x-y plane may be specified 16 so that the new x axis is the major axis of the data.

Since the patch edge projections on the x-y plane are in general curved, it is necessary to extend the original surface 17 so that any rectangles to be chosen, which lie partly within the original projected patches, lie entirely within the extended surface.

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