I’m saying that the tangent of a straight line in Cartesian coordinates, projected into polar, does not have constant tangent. A line with a constant tangent in polar, would look like a circle in Cartesian.
We are interested in the lines tangent a given graph, regardless of whether that graph is produced by rectangular, parametric, or polar equations. In each of these contexts, the slope
of the tangent line is dydx. Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ. Instead, we will use x=f(θ)cosθ, y=f(θ)sinθ to compute dydx.
From the link above. I really don’t understand why you seem to think a tangent line in polar coordinates would be a circle.
Define straight in a precise, mathematical way.
The tangent of all points along the line equal that line
Only true in Cartesian coordinates.
A straight line in polar coordinates with the same tangent would be a circle.
EDIT: it is still a “straight” line. But then the result of a square on a surface is not the same shape any more.
I’m not sure that’s true. In non-euclidean geometry it might be, but aren’t polar coordinates just an alternative way of expressing cartesian?
Looking at a libre textbook, it seems to be showing that a tangent line in polar coordinates is still a straight line, not a circle.
I’m saying that the tangent of a straight line in Cartesian coordinates, projected into polar, does not have constant tangent. A line with a constant tangent in polar, would look like a circle in Cartesian.
From the link above. I really don’t understand why you seem to think a tangent line in polar coordinates would be a circle.
Sorry that’s not what I’m saying.
I’m saying a line with constant tangent would be a circle not a line.
Let me try another way, a function with constant first derivative in polar coordinates, would draw a circle in Cartesian
I think this part from the textbook describes what you’re talking about
And this would give you the actual tangent line, or at least the slope of that line.
But then your definition of a straight line produces two different shapes.
Starting with the same definition of straight for both. Y(x) such that y’(x) = C produces a function of cx+b.
This produces a line
However if we have the radius r as a function of a (sorry I’m on my phone and don’t have a Greek keyboard).
R(a) such that r’(a)=C produces ra +d
However that produces a circle, not a line.
So your definition of straight isn’t true in general.
geodesic