CSL Notebook Entry: Some properties of Interpress Conic Curves

XEROX
To    From
CSL    Dennis Arnon
    PARC
Subject        Date
Interpress Conic Curves    January 28, 1987

<<Copyright © 1987 by Xerox Corporation.  All rights reserved.>>
FOR XEROX INTERNAL USE ONLY

Abstract    Pieces of conic curves in Interpress 3.0 are defined by a scheme that goes back at least to A. Newell at Boeing (1960). 
 A conic patch is specified by two endpoints P and R, and a point T of intersection with segment [Q, S], where Q is a third point 
and S is the midpoint of [P, R]. The following figure illustrates the scheme:
<< [Artwork node; type 'ArtworkInterpress on' to command tool] >>
We derive solutions to two problems pertaining to such conics.  First, suppose the conic is a (piece of a) parabola, i.e. T is the 
midpoint of [Q, S]; then find the point U of [Q, S] such that triangle [P, U, R] is a best approximation to the parabola (in a 
sense to be made precise below). Second, for any conic patch defined by this scheme, find the implicit equation of the conic curve 
of which the patch is a piece.
Attributes    informal, technical, Page imaging, Splines
Introduction
    The solution to the first problem is used by the Imager to render conics.  A given conic is subdivided until one has pieces that 
    can be sufficiently well approximated as parabolas, then the area-difference-minimizing triangle of each such parabolic patch 
    is rendered.

The second problem is just for fun.

Historical note: Robin Forrest's Ph. D. thesis ([FOR68], p. 30) cites Newell [NEW60] for some basic facts about this conic 
definition scheme.  Hence it must go back at least that far.
First problem 
It suffices to solve the problem for a particular parabola.  Hence we use the simple example suggested by Figure 1:

<< [Artwork node; type 'ArtworkInterpress on' to command tool] >>
Figure 1
The equation of this parabola is 
                     Y .
Figure 2 adds a triangle and appropriate notation to Figure 1:
<< [Artwork node; type 'ArtworkInterpress on' to command tool] >>
Figure 2
Let X  be the equation of the line through R and U. We wish to choose U such that
                  Y

is minimized.  Call the U having this property minimal, and similarly triangle [R, S, U] for this U is the minimal triangle. 
Clearly any U below T or above Q is not minimal, and for minimal U, line [R, U] intersects the parabola in a point
I =  (X, X),
with X in the interval  Y of the x-axis.  So let us solve for the X corresponding to minimal U.
The equation of the line through R and U is
Y
For x < X, the line is above the parabola.  For x > X, the parabola is above the line. Hence the function we want to minimize is:
Y

Routine computation reveals that:
Y
This function has a local minimum at
Y
Computation of Y and Y reveals that
Y
is an absolute minimum on the interval of interest, yielding minimal U of:
Y
It is convenient to express the ratio [U, S] / [Q, S], which is:
X
Second problem
The general implicit equation of a conic curve can be written:
Y.
We have five constraints on a conic patch defined by the Interpress scheme:
(1) It must pass through P. 
(2) It must pass through R. 
(3) Its slope at P must be the slope of line [P, Q]. 
(4) Its slope at R must be the slope of line [R, Q]. 
(5) It must pass through T. 
Clearly constraints (1), (2), and (5) give rise to a homogeneous linear equations in a, b, c, d, e, and f.  By the Implicit 
Function Theorem, the slope of any curve X at a point X on the curve is given by
  X
where X and  X  are the respective partial derivatives of  X.   Hence constraints (3) and (4) also give rise to homogeneous linear 
equations in a, b, c, d, e, and f. Solving the resulting 5 by 6 system of homogeneous linear equations gives us a, b, c, d, e, and 
f up to a constant multiple.
Let's go through this for the particular parabola we considered in the first problem.  The constraints give us equations:  
(1) a - e + f = 0. 
(2) a + e + f = 0.  
(3) 2a + 2b -2d - e = 0. 
(4) 2a - 2b -2d + e = 0. 
(5) c + d + f = 0.
Hence the system to solve is:
      X
which yields an equation
X
for this parabola (where X is the arbitrary constant multiple noted above).
Acknowledgements
Thanks to Dan Bloomberg and Eric Bier for corrections and comments on an earlier draft.
References
[FOR68]
R Forrest, Curves and surfaces for computer-aided design, Cambridge University, 1968.
[NEW60]
A Newell, A general discussion of the use of conic equations to define curved surfaces, The Boeing Company, Document D2-4398, March 
1960.