Parametric Curves and Surfaces
(Page: 1
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 3 ) Text CONTINUITY OF PARAMETRIC CURVES
A parametric curve will be continuous if all of its component functions are continuous.
For parametric curves in 2space, this means that the curve will be continuous if the following condition is met: For some arbitrarily small ε_{x} and ε_{y} and for every value t_{0} of t, it is always possible to choose a &delta greater than 0 but small enough that for the part of the curve such that t_{0}  &delta ≤ t ≤ t_{0} + &delta, x(t_{0})  ε_{x} ≤ x(t) ≤ x(t_{0}) + ε_{x} and y(t_{0})  ε_{y} ≤ y(t) ≤ y(t_{0}) + ε_{y}.
A parametric curve in 3space will be continuous if the following condition is met:
For some arbitrarily small ε_{x}, ε_{y}, and ε_{z} and for every value t_{0} of t, it is always possible to choose a &delta greater than 0 but small enough that for the part of the curve such that t_{0}  &delta ≤ t ≤ t_{0} + &delta, x(t_{0})  ε_{x} ≤ x(t) ≤ x(t_{0}) + ε_{x}, y(t_{0})  ε_{y} ≤ y(t) ≤ y(t_{0}) + ε_{y}, and z(t_{0})  ε_{z} ≤ z(t) ≤ z(t_{0}) + ε_{z}.
Demos
Continuity of Parametric Curves in 2Space
 
See if for arbitrary small epsilonX and epsilonY, it is possible to choose a nonzero delta small enough that the magenta part of curve lies inside the yellow rectangular box for certain values t0 of t, particularly those where there appears to be a discontinuity (if there are such values). If the xcomponent of the parametric curve is continuous, it will be possible to get the magenta part to lie between the red lines given a small enough delta. If the ycomponent is continous, it will be possible to get the magenta part to lie between the blue lines. If both parts are continous, the parametric curve will be continous and it will be possible to fit the magenta part of the curve inside the box.
Note for extremely small numbers (on the order of 10^{16}) the application will round to 0.

Continuity of Parametric Curves in 3Space
 
This demo resembles the one above, with the principal difference here being the presence of an additional coordinate (z). See if, given an arbitrarily small box and some value t0 of t, it is possible to find a nonzero delta small enough to fit the magenta part of the curve inside the box.

