Why Circle is traced counterclockwise and ellipse is traced clock wise?

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For one curve/surface, we can parametrize it differently. For example, consider an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. It can be parametrized as $\langle a\cos(t),b\sin(t)\rangle$, or as $\langle a\cos(t),-b\sin(t)\rangle$. You can verify that the first parametrization is counter-clockwise, and the second one is clockwise.

There is no principle, per se, in choosing parametrization as long as it is correct. However, sometimes we may want to choose a specific type of parametrization for various reasons. One simple example is the arc-length parametrization. If you parametrize a curve using an arc-length parametrization, the speed (the norm of velocity) will always equal to $1$, which makes life a bit easier in some cases. Another example is that, in order to use some theorems (such as Stoke's Theorem), the theorems require that the curve is parametrized in counter-clockwise direction.

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Updated on July 18, 2020

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  • Community
    Community over 3 years

    In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise.

    "Which was this one here. And first we noted that this does parameterize, as we say, the circle. That satisfies the equation for the circle. And it's traced counterclockwise."

    "So this is what happens at $ t = 0 $. This is where we are at $ t = \frac{\pi}{2} $. And it continues all the way around, etc. To the rest of the ellipse. This is the direction. So this one happens to be clockwise."

    What is the principle behind choosing the direction of tracing the curves?

    • 3x89g2
      3x89g2 over 7 years
      It might not be a principle, but sometimes the parametrization is required to be traced counterclockwise in order to perform some integrals. For example, Stoke's Theorem.
    • Admin
      Admin over 7 years
      So,there is not a definite rule?
    • 3x89g2
      3x89g2 over 7 years
      I am not sure what you are trying to ask. For one curve/surface, there are many different ways to parametrize it and they are all valid. We use different parametrization for different purposes.
    • Admin
      Admin over 7 years
      For example,can we trace an ellipse in both clockwise and counter-clockwise direction?
    • 3x89g2
      3x89g2 over 7 years
      I am pretty sure you can.