Can Smooth Curves in 3D Have Cusps?

In summary, the conversation discusses the concept of "smooth" curves in 3D and how differentiation can turn cusps into discontinuities. However, the curve (1+t^2,t^2,t^3) has a cusp at (1,0,0) but its derivative (2t,2t,3t^2) is smooth. The explanation for this lies in the frame of reference and the condition that R'(t) ≠ 0.
  • #1
inkliing
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"smooth" curves with cusps in 3d

While reviewing basic calculus, I noticed that the curve (1+t^2,t^2,t^3), which clearly has a cusp at (1,0,0), has a derivative curve (2t,2t,3t^2) which is clearly smooth. This struck me as odd since differentiation usually seems to turn cusps into discontinuities, whereas integration smoothes out a curve, especially a curve described by polynomials. In fact, in general I have always taken a curve to be smooth iff it has a continuous derivative, which this curve has, and yet a cusp cannot be smooth in any sensible sense. I suspect the explanation is relatively simple - just something I'm missing.

Thx in advance.
 
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  • #2
hi inkliing! :smile:
inkliing said:
… differentiation usually seems to turn cusps into discontinuities …

no, that's only for cusps that have a non-zero angle

a cusp with a zero angle is often an illusion

consider a point on the wheel of a steadily moving car …

in the frame of reference of the car, it's going in a uniform circle (you can't get any smoother than that!), with https://www.physicsforums.com/library.php?do=view_item&itemid=27" of constant magnitude v2/r

but in the frame of reference of the ground, it follows a cycloid (see http://en.wikipedia.org/wiki/Cycloid" for a neat .gif), with a cusp whenever that point contacts the ground …

it moves vertically down just before contact, and vertically up just after …

but it still obviously has acceleration of constant magnitude v2/r :wink:

(can you find a frame of reference in which your curve has no cusp? :biggrin:)
 
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  • #3


Note also that your "cusp" is at t= 0 where the derivative is (0, 0, 0) so that is NOT a proper parameterization of the curve.
 
  • #4


Thx tiny tim for the very straightforward frame-of-reference refrence. I understand it much better now :)
 
  • #5


inkliing said:
While reviewing basic calculus, I noticed that the curve (1+t^2,t^2,t^3), which clearly has a cusp at (1,0,0), has a derivative curve (2t,2t,3t^2) which is clearly smooth. This struck me as odd since differentiation usually seems to turn cusps into discontinuities, whereas integration smoothes out a curve, especially a curve described by polynomials. In fact, in general I have always taken a curve to be smooth iff it has a continuous derivative, which this curve has, and yet a cusp cannot be smooth in any sensible sense. I suspect the explanation is relatively simple - just something I'm missing.

Thx in advance.

The condition you need to avoid such "smooth" cusps is that R'(t) ≠ 0. If think of an object moving, if you allow it to smoothly come to a stop then smoothly take off in a different direction, you can get sharp corners. But if you have a continuous non-zero derivative for R(t), that can't happen, and that is the definition of a smooth parametric curve.
 

Related to Can Smooth Curves in 3D Have Cusps?

1. What are smooth curves with cusps in 3D?

Smooth curves with cusps in 3D are mathematical curves that have a smooth and continuous shape, but have a sharp point or corner called a cusp. These curves can be described using parametric equations and are commonly found in 3D modeling and computer graphics.

2. How do you identify a cusp in a 3D curve?

A cusp in a 3D curve can be identified by looking for a sharp point or corner in the curve's shape. Mathematically, a cusp can also be identified by a singularity in the curve's derivative, where the slope changes suddenly.

3. What causes a cusp in a 3D curve?

A cusp in a 3D curve is caused by a change in the curve's direction or curvature. This can occur when two branches of the curve meet at a sharp angle, resulting in a cusp point. In some cases, cusps can also be intentionally created to add detail and complexity to a 3D model.

4. How can smooth curves with cusps be used in 3D modeling?

Smooth curves with cusps can be used in 3D modeling to add more realistic and detailed shapes to a model. They can also be used to create sharper edges and corners, which are common in real-world objects. Additionally, these curves can be used in animation to create more dynamic and lifelike movements.

5. Are there any real-life examples of smooth curves with cusps in 3D?

Yes, there are many real-life examples of smooth curves with cusps in 3D. Some common examples include the sharp edges of a pyramid or cone, the curves of a seashell, and the shape of a rooster's comb. These curves can also be found in the design of buildings, furniture, and other man-made objects.

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