tech.pl

#!/usr/bin/perl

use TEMPL;
TEMPL::Init();

$TEMPL::TITLE = 'SolveSpace - Technology';

TEMPL::OutputWithHeader("TECHNOLOGY: SOLVING CONSTRAINTS", <<EOT

<p>The core of any parametric CAD program is its geometric constraint
solver. The solver starts with geometric properties of the sketch,
like a line length or an angle or a tangency. Based on these properties,
it calculates a simple representation of the sketch's lines, points, and
curves.</p>

<p>For a trivial example, we might specify that a point lies 10 mm
from the origin, and 5 mm from the x-axis. We also specify that the
line from the origin to that point makes a 65&deg; angle with the
z-axis. That point turns out to have (x, y, z) = (8.66, 2.67, 4.23),
after solving three equations in three unknowns. A real sketch may involve
hundreds of unknowns.</p>

<div class="forimg">
    <img src="pics/tech-example.png">
</div>

<p>In SolveSpace, constraints are represented as equations in a
symbolic algebra system. In general, these equations are solved
numerically, by a modified Newton's method. Some special cases are
handled, for any equation that can be solved by forward-substitution,
and equations that are solvable alone.</p>

<p>If the sketch is not fully constrained, then the Jacobian is solved
in a least squares sense, with each equation written in such a way as
to make the quantity being minimized a useful penalty metric. This tends
to yield "less surprising" results when dragging points in an underconstrained
sketch, compared to approaches that use weak dimensions to keep the
sketch always fully constrained.</p>

<p>A failure to converge almost always results from a real problem
with the specified constraints. For example, a sketch that describes
a linkage will fail to converge when the real linkage would bind. If
multiple solutions exist, then the initial position of the sketch
determines which is chosen. The natural result (according to the
basin of attraction of the numerical method) is generally a reasonable
choice, though some cases are handled specially.</p>

<p>In general, constraints apply in 3d. A constraint may also apply
projected into a workplane. For example, a constraint on the distance
between two points may actually apply to the distance between the
projections of those two points into the specified plane. Angle,
point-coincident, and point-on-curve constraints are similarly available
in projected forms.</p>

<p>The constraint solver used in SolveSpace is available separately
as a <a href="library.$TEMPL::PL">library</a>.</p>

EOT
);