2.4. REGULAR CURVES185component functions has nonzero derivative on an interval of the form|t−t0|<εforε>0sufficiently small; if the first component has thisproperty, then the projection of the subcurve defined by this inequality ontothexy-plane (resp.xz-plane) is the graph ofy(resp. ofz) as aC1function ofx.From this we can conclude that, as in the planar case, the tangent line tothe parametrization at any particular parameter valuet=t0iswell-defined, and is the line in space going through the point−→p(t0) withdirection vector−→v(t0); furthermore, the linearization of−→p(t) att=t0is aregular vector-valued function which parametrizes this line, and hasfirst-order contact with−→p(t) att=t0.As a quick example, we consider the vector-valued function−→p(t) = (cost,sint,sin 3t)with velocity−→v(t) = (−sint,cost,3 cos 3t).Since sintand costcannot both be zero at the same time, this is a regularparametrization of a curve in space, sketched in Figure2.26. We note, for
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