Finding a Piecewise Smooth Parametric Curve for the Astroid

In summary, the conversation discusses finding a piecewise smooth parametric curve for the astroid given by the equation $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$. The individuals in the conversation discuss different attempts at solving this problem and questioning the possibility of a piecewise smooth curve. They also discuss the use of phi and theta as variables and the potential confusion it may cause.
  • #1
ttsp
6
0

Homework Statement


Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve?

Homework Equations


$\phi(\theta) = (cos^3(\theta),sin^3(\theta))$

The Attempt at a Solution


I have tried to use the cartesian equation x^(2/3) + y^(2/3) = 1 but that didn't help. I tried to change the periodicity of the cos and sine functions but obviously that was pointless. I thought at one time maybe this is not possible, but I see examples of line integrals over the astroid so it must be piecewise smooth. Can I get a hint?
 
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  • #2
How can phi be a vector?

Just by looking at the plot of the curve, which pieces seem useful?
 
  • #3
ttsp said:

Homework Statement


Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve?

Homework Equations


##\phi(\theta) = (cos^3(\theta),sin^3(\theta))##
That is poorly written, even after fixing the tex as I did. Using phi and theta as your variables suggests spherical coordinates somehow. Let's rewrite it as ##\vec R(t) = \langle \cos^3(t),\sin^3(t) \rangle##. Now I am confused. That equation gives a graph that is piecewise smooth already, doesn't it? Or is your problem really to find the points of the vector function where it violates the definition of "smooth", which is not the same thing as the graph having sharp corners?
 
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Related to Finding a Piecewise Smooth Parametric Curve for the Astroid

1. What is a piecewise smooth parametric curve?

A piecewise smooth parametric curve is a curve that is defined by multiple smooth parametric equations, each defined on a specific interval. These equations are "pieced" together to form a continuous curve.

2. What is an astroid?

An astroid is a specific type of curve known as a hypocycloid. It is defined by the parametric equations x = a*cos^3(t) and y = a*sin^3(t), where a is a constant. It is also known as a "star curve" because of its resemblance to a star.

3. Why is it important to find a piecewise smooth parametric curve for the astroid?

Finding a piecewise smooth parametric curve for the astroid allows us to better understand and visualize the curve. It also allows us to more easily calculate important properties of the curve, such as its length and area.

4. What is the process for finding a piecewise smooth parametric curve for the astroid?

The process involves breaking the astroid curve into smaller sections and finding a smooth parametric equation for each section. This can be done by using mathematical techniques such as trigonometric identities and calculus. The equations for each section are then "pieced" together to form a continuous curve.

5. Are there any real-world applications for the astroid curve?

Yes, the astroid curve has been used in various engineering and design fields, such as construction, architecture, and even in the design of gears. It is also commonly used in mathematical visualizations and animations.

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