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346 15. Curves
Some properties of curves are attributed to only a single location on the curve,
while other properties require knowledge of the whole curve. For an intuition of
the difference, imagine that the curve is a train track. If you are standing on the
track on a foggy day you can tell that the track is straight or curved and whether
or not you are at an end point. These are lo cal properties. You cannot tell whether
or not the track is a closed curve, or crosses itself, or how long it is. We call this
type of property, a global property.
The study of local properties of geometric objects (curves and surfaces) is
known as differential geometry. Technically, to be a differential property, there
are some mathematical restrictions about the properties (roughly speaking, in the
train-track analogy, you would not be able to have a GPS or a compass). Rather
than worry about this distinction, we will use the term local property rather than
differential property.
Local properties are important tools for describing curves because they do not
require knowledge about the whole curve. Local properties include
• continuity,
• position at a specificplaceonthecurve,
• direction at a specificplaceonthecurve,
• curvature (and other derivatives).
Often, we want to specify that a curve includes a particular point. A curve is
said to interpolate a point if that point is part of the curve. A function f interpo-
lates a value v if there is some value of the parameter u for which f (t)=v. We
call the place of interpolation, that is the value of t, the site.
15.2.1 Continuity
It will be very important to understand the local properties of a curve where two
parametric pieces come together. If a curve is defined using an equation like
Equation (15.2), then we need to be careful about how the pieces are defined. If
f
1
(1) = f
2
(0), then the curve will be “broken”—we would not be able to draw
the curve in a continuous stroke of a pen. We call the condition that the curve
pieces fit together continuity conditions because if they hold, the curve can be
drawn as a continuous piece. Because our definition of ”curve” at the beginning
of the chapter requires a curve to be continuous, technically a ”broken curve” is
not a curve.