# The speed of a curve in different coordinate systems

• kkz23691
In summary, the conversation discusses the concept of unit speed curves in different coordinate systems and whether the speed of a curve changes depending on the coordinates used. It is noted that the length of the tangent vector to the curve at any point is an intrinsic property of the curve and does not depend on the coordinate system. However, it is also mentioned that the second derivative of the curve may vary depending on the coordinate system. The term "unit speed curve" is clarified to be equivalent to a curve parameterized by arc length, where the derivative with respect to the parameter is a unit-length tangent vector.
kkz23691
Hello,

If for a curve in Cartesian coordinates ##||\dot{{\mathbf r}}||=\mbox{const}## (i.e. the curve is constant speed) will the speed of the curve change in cylindrical and spherical coordinates?
Could someone experienced share how the transition from flat Euclidian space to curved space alters the speed of the curve?

Thanks!

I am puzzled by the expression "speed of a curve". A curve just sits there! It doesn't have any "speed". But I assume you mean the length of the tangent vector to the curve at any point which could also be thought of as the speed of an object moving along the curve. That is an "intrinsic" property of the curve, it does not depend on the coordinate system.

kkz23691
HallsofIvy, I hear you :) "Unit speed" seems to be a term used in math.
It turns out, ##||\ddot{{\mathbf r}}||## depends on the coordinate system; it wasn't obvious to me that ##||\dot{{\mathbf r}}||## doesn't.

To add this - even in the same coordinate system a curve can be reparametrized from a non-unit-speed to unit-speed. If a curve is already unit-speed in some coordinate system, I am uncertain why it would still be unit-speed in any other coordinate system.

kkz23691 said:
HallsofIvy, I hear you :) "Unit speed" seems to be a term used in math.
It turns out, ##||\ddot{{\mathbf r}}||## depends on the coordinate system; it wasn't obvious to me that ##||\dot{{\mathbf r}}||## doesn't.
I would be very surprised if "speed", a physics term, were used in math. What you are calling a "unit speed curve" seems to be, mathematically, a "curve parameterized by arc length". Such an expression for a curve always has the property that the derivative (x, y, z), the "direction vector" of the curve, with respect to the parameter, arclength, is a unit-length tangent vector. If you think of the parameter as "time", t, the that derivative is the velocity vector but that is again a "physics", not "math" concept.

## 1. What is the speed of a curve in different coordinate systems?

The speed of a curve is a measure of how fast it is moving at a given point. In different coordinate systems, the speed of a curve can vary depending on the orientation and scale of the axes.

## 2. How is the speed of a curve calculated?

The speed of a curve is typically calculated using calculus, specifically the derivative of the curve's position function. This gives the instantaneous rate of change, or velocity, at a given point on the curve.

## 3. Can the speed of a curve be negative?

Yes, the speed of a curve can be negative. This indicates that the curve is moving in the opposite direction of the positive direction of the axis. For example, a car moving backwards on a straight road would have a negative speed.

## 4. How does the speed of a curve change in different coordinate systems?

The speed of a curve can change in different coordinate systems because the orientation and scale of the axes can affect the calculation of the derivative. Additionally, different coordinate systems may have different units of measurement, which can also impact the speed of the curve.

## 5. What is the significance of understanding the speed of a curve in different coordinate systems?

Understanding the speed of a curve in different coordinate systems is important in many scientific fields, such as physics and engineering. It allows for accurate calculations and predictions of motion, and can also provide insights into the relationship between different coordinate systems and their effects on the speed of a curve.

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