How does regularity of curves prevent "cusps"?

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In summary: At a kink, the direction of the tangent vector changes abruptly, which means that the velocity of the point tracing the curve also changes abruptly. This kind of sudden change in velocity is only possible if ##\dot\gamma(t)## is zero at the kink. Similarly, at a cusp, the curve changes direction very sharply, which again requires ##\dot\gamma(t)## to be zero. Therefore, forbidding ##\dot\gamma(t) = 0## ensures that there are no cusps or kinks in the curve.
  • #1
center o bass
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A regular curve on a manifold ##M## is a curve ##\gamma:I \to M## such that ##\dot \gamma(t) \neq 0## for any ##t \in I##. In John Lee's "Introduction to Curvature" he says that this intuitively means that we prevent the curve from having "cusps" and "kinks".

How can I see that this is the case? I.e. how could ##\dot \gamma(t) = 0## for some t lead to a "kink" or "cusp"?
 
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  • #2
center o bass said:
A regular curve on a manifold ##M## is a curve ##\gamma:I \to M## such that ##\dot \gamma(t) \neq 0## for any ##t \in I##. In John Lee's "Introduction to Curvature" he says that this intuitively means that we prevent the curve from having "cusps" and "kinks".

How can I see that this is the case? I.e. how could ##\dot \gamma(t) = 0## for some t lead to a "kink" or "cusp"?

It doesn't. ##\dot \gamma(t) = 0## means a stationary point and can happen just from parametrization. But cusps or kinks do require ##\dot \gamma(t) = 0##, so forbidding it ensures you get none.

##\dot \gamma(t) \neq 0## is needed for two other reasons : it ensures ##\dot \gamma(t) ## gives you the direction of the tangent vector to the curve at every point, and that the parametrization is locally one-to-one / doesn't backtrack.
 
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  • #3
wabbit said:
It doesn't. ##\dot \gamma(t) = 0## means a stationary point and can happen just from parametrization. But cusps or kinks do require ##\dot \gamma(t) = 0##, so forbidding it ensures you get none.

##\dot \gamma(t) \neq 0## is needed for two other reasons : it ensures ##\dot \gamma(t) ## gives you the direction of the tangent vector to the curve at every point, and that the parametrization is locally one-to-one / doesn't backtrack.
But why -- intuitively -- do cusps or kinks require ##\dot gamma = 0##?
 
  • #4
One way to see it is to think of the curve as traced by a moving point at ## \gamma(t) ##; then ## \dot\gamma(t) ## is the velocity of that point. If it's not zero it has a direction. What is that direction at a kink ?
 

1. What is a "cusp" in terms of curves?

A "cusp" is a point on a curve where there is a sharp change in the direction of the curve, resulting in a pointy corner or tip. It is also known as a "singularity" or "vertex".

2. How does the regularity of curves prevent cusps?

The regularity of curves refers to the smoothness and continuity of the curve. A curve that is regular does not have sharp changes in direction or sudden jumps, which are necessary conditions for a cusp to occur. Therefore, the more regular a curve is, the less likely it is to have cusps.

3. What factors contribute to the regularity of curves?

There are several factors that contribute to the regularity of curves, including the degree of the curve (higher degree curves are generally more regular), the number of control points or data points used to define the curve, and the type of curve (e.g. linear, quadratic, cubic, etc.). Additionally, the mathematical algorithm used to construct the curve can also impact its regularity.

4. Can curves with cusps be considered regular?

No, according to mathematical definitions, curves with cusps are not considered regular because they do not meet the criteria for smoothness and continuity. However, in certain applications, such as computer graphics, curves with cusps may still be used and may be considered "regular enough" for the desired purpose.

5. Are there any benefits to having cusps in curves?

In some cases, cusps can be intentionally created in curves to add points of interest or to create sharp angles. For example, in art and design, cusps can be used to create more visually appealing and dynamic shapes. Additionally, in certain mathematical and scientific applications, cusps can provide critical information about the behavior and properties of the curve.

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