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Line Bicubic Spline Intersections

IP.com Disclosure Number: IPCOM000077995D
Original Publication Date: 1972-Oct-01
Included in the Prior Art Database: 2005-Feb-25
Document File: 3 page(s) / 57K

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Dimsdale, B: AUTHOR


Consider the line x = x(O) + alphat y = y(O) + betat z = z(O) + gammat and a surface

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Line Bicubic Spline Intersections

Consider the line x = x(O) + alphat y = y(O) + betat z = z(O) + gammat and a surface

z = B(x,y) which is bicubic in x and y and defined over a rectangel x(i)</=/</=/x(i)+1, y(j)</=/y<(j+1. Suppose it is known also that ax + by + c(2) </=/z</=/ ax + by c(1). that is, the bicubic segment is contained in a box (a parallelepiped) consisting of four vertical planes and two slant planes. Then either the line intersects two faces of the box, determining values t(1),t(2) such that the line is inside the box for t(2)</=/t</=/t(1), or the line intersects no faces of the box. In the latter case the line does not intersect the surface segment. In the former case, any values of t for which the line intersects the surface must lie between t(1) and t(2). Furthermore, phi(t) = B(x(0)+apha t, y()) + Beta t) -z(0) -Grammar t. is a sixth order polynomial which measures the vertical distance between a point on the line and the surface, and certainly bounds the perpendicular distance from the line to the surface. When phi (t) = 0 the line and surface intersect. In practice, there exists an Epsilon such that phi absolute (t) </=/Epsilon defines intersection for practical purposes. The following is a discussion of the problem phi(t) = 0, thus ignoring a few complexities.

Since this problem is at the heart of APT processing for machine tool applications, speed is the essential requirement. The strategy of the flow diagram used to obtain maximum speed is to eliminate cases as rapidly as possible, while guaranteeing at the same time that no true possibility of a solution is overlooked. Gross input for the problem consists of the numbers x(1), x(2), . .
. , x(N)

y(1), y(2), . . . , y(M). which are monotone increasing and define the grid lines. Also A, B, C(1), C(2) are given such that Ax + By + C(2) </=/ z(x,y) </=/ Ax + By + C(1). over the domain of definition of the surface. Also Gam...