# Can Smooth Curves in 3D Have Cusps?

• inkliing
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.
inkliing
"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.

hi inkliing!
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

(can you find a frame of reference in which your curve has no cusp? )

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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.

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

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.

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.

## 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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