I have a question in introductory differential geometry: if I know that a regular curve in the plane c(t) has a constant magnitude R 0, for all t in I, the domain. How would I show that c(t) is a partial reparametrization of a circle? Also, how would I show that the curvature of c(t) is 1/R, simply from the condition given above?

$$c: I \rightarrow \mathbb{R}^2, \ \parallel c \parallel = R, \ R 0$$

The definition of a partial reparametrization of a curve is as follows: If $$c: I \rightarrow \mathbb{R}^2$$ is a regular curve and $$\phi : I' \rightarrow I$$ is a smooth and increasing injection, then $$ c \circ \phi $$ is a partial reparametrization of the curve $$c$$.

I'm not really sure how to show the first part explicitly without making a hand-wavy argument about a smooth injection from I into the domain of the parametrized form of a circle, which is $$ [0, 2\pi ] $$

Question

I have a question in introductory differential geometry: if I know that a regular curve in the plane c(t) has a constant magnitude R > 0, for all t in I, the domain. How would I show that c(t) is a partial reparametrization of a circle? Also, how would I show that the curvature of c(t) is 1/R, simply from the condition given above?

$$c: I \rightarrow \mathbb{R}^2, \ \parallel c \parallel = R, \ R > 0$$

The definition of a partial reparametrization of a curve is as follows: If $$c: I \rightarrow \mathbb{R}^2$$ is a regular curve and $$\phi : I' \rightarrow I$$ is a smooth and increasing injection, then $$ c \circ \phi $$ is a partial reparametrization of the curve $$c$$.

I'm not really sure how to show the first part explicitly without making a hand-wavy argument about a smooth injection from I into the domain of the parametrized form of a circle, which is $$ [0, 2\pi ] $$

anonymous · Answer

What is $I'$?

dan815 · Answer

so is it illegal to use differential calculus??
parametrizing circle c(t) and calculating dT/ds change in unit tangent wrt to arc length