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CHAPTER 6 CONTOURS
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Figure 68: Conic sections are defined by intersecting a cone with a plane end, with equal tangents at the knots to provide a smooth transition between adjacent sections of the curve Let the polyline vertices be Vi The conic approximation is shown in Figure 69 Each conic section in a conic spline is defined by two end points, two tangents, and one additional point The knots Ki can be located between the vertices of the polyline:
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(625)
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where Vi is between 0 and 1 The tangents are defined by the triangle with vertices Vi, Vi+! , and Vi+2 The additional point is
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(626)
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as shown in Figure 610 There are several special cases of the conic section that can be handled in a uniform way by this representation If Vi+l = 0, then the ith section of the conic spline is the line segment from Ki to Vi+l If Vi = 1 and Vi+l = 0, then K i , Ki+ 1, and Vi+! collapse to the same point and there is a corner in the sequence of conic sections These special properties allow line segments and corners to be represented explicitly in a conic spline, without resorting to different yrimitives or special flags The algorithm presented here for computing conic splines uses the guided form of a conic section, which represents a conic section using three lines that
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66 CONIC SECTIONS
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Figure 69: Conic sections are approximations defined between three points
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Figure 610: A conic section is defined by the two end points and tangents obtained from three vertices of the polyline approximation, plus one additional point
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CHAPTER 6 CONTOURS
Figure 611: The guided form for a conic bound the conic (See Figure 611) The equation of a line is
(627)
Let the first and last vertices in a polyline be A and B, and let point C be an intermediate vertex in the polyline The first and last vertices are joined by the chord AB The guided form of conic is the family of conics with end points at A and B and tangents AC and BC defined by the equation
(628)
where
+ alx + a2Y =
(629)
is the line containing the line segment AC, (630) is the line containing the line segment BC, and (631)
is the line containing the chord AB The family of conic sections is parameterized by p
67 SPLINE CURVES
The algorithm for fitting a conic section to a list of edge points starts with a polyline and classifies the vertices as corners, soft vertices, or knots Soft vertices have angles near 180 0 , and the adjacent line segments are nearly collinear and may be replaced with a conic section A sequence of soft vertices corresponds to a sequence of line segments with gradually changing orientation that most likely were fitted to edge points sampled along a smooth curve Corners have vertex angles above 180 0 + Tl or below 180 0 - T 1 , where Tl is a threshold, and are unlikely to be part of the conic Knots are placed along a line segment that has soft vertices at either end that are angled in opposite directions A conic section cannot have an inflection, so two conic sections must be joined at the knot The placement of the knot along the line segment is determined by the relative angles of the soft vertices at the ends of the line segment Let the angles of the two soft vertices Vi and Vi+! be Ai and Ai+l' respectively If Ai = Ai+l' then the knot is placed halfway between the vertices, which means that v = 1/2 in Equation 625 If the angles are not the same, then the knot location should be biased away from the vertex with the larger angle, since the conic may not bend away from the line segment fast enough to follow the corner The value for v in Equation 625 can be set using the formula (632) Each sequence of line segments joined by soft vertices is replaced by a guided conic through the first and last vertices (or knots) The tangents are defined by the orientation of the first and last line segments The tangents and end points determine four of the five degrees of freedom for the conic The conic is fully specified by having it pass through the soft vertex in the middle of the sequence
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