Curve analysis

Theory

In the parametric representation, a curve is a continuous mapping from the scalar interval [Umin, Umax] to points in space.

This means that every point P on the curve has a corresponding scalar parameter U. The length of a curve segment generally differs from the difference between its endpoint parameters.

This section describes methods relating curve parameters to points and lengths.

Curve classes

ZenCad provides curve analysis methods in the following classes:

Endpoints and parameter range

Finding the endpoints of finite curves.

The endpoints method returns the endpoint objects. The range method returns their parameters.

curve.endpoints() # -> tuple[Point3, Point3]
curve.range() # -> Interval; .lower/.upper -> Scalar

crv = circle(r=5, wire=True, angle=deg(270))
s,f = crv.endpoints()
disp([crv, s, f])

curve.d0(u)

Returns the point at parameter u.

curve.d1(u)

Returns the first derivative vector at parameter u.

curve.lower_distance_parameter(pnt)

Returns the parameter of the point on the curve closest to pnt.

Equally spaced points

Returns an array of points equally spaced along the curve. The npnts parameter sets the number of points. It must be an integer of at least two. Specify both range boundaries or omit both; the result includes the endpoints. Spacing is measured along the curve, rather than in parameter space. The umin and umax parameters define the parameter range to distribute the points over.

curve.uniform(npnts, U_min, U_max) # -> list[Scalar]
curve.uniform_points(npnts, U_min, U_max) # -> list[Point3]

crv = circle(r=5, wire=True, angle=deg(270))

params = crv.uniform(8, math.pi/4, math.pi)
print([float(p) for p in params]) # [0.7853981633974483, 1.121997376282069, 1.4585965891666897, 1.7951958020513104, 2.131795014935931, 2.4683942278205517, 2.8049934407051724, 3.141592653589793]

pnts = crv.uniform_points(8, math.pi/4, math.pi)
disp(pnts + [crv])

Two-dimensional curves

Curve2 has its own interface: point(u) returns a Point2, tangent(u) returns a Vector2 of the first derivative, and range() returns an Interval. Use trim(start, end) to restrict the range. The d0, d1, endpoints and uniform_points methods described above for Edge and Curve are not part of the Curve2 interface.

from zencad import *

curve2 = segment2(point2(0, 0), point2(10, 0))
interval = curve2.range()
start = curve2.point(interval.lower)
end = curve2.point(interval.upper)
assert float(start.x) == 0
assert float(end.x) == 10
assert isinstance(curve2.tangent(interval.lower), Vector2)