Suppose we have such a shape:
It is needed to found what this shape's edges are.
I mean, this shape edges are:
- outer arc (upper) - we know everything we would like about this arc: radius, start and end angles, start and end points, center point – everything.
- bottom straight line - we know everything we would like about this line too: start and end point
- right straight line - we know everything we would like about this line too: start and end point.
- inner arc (lower) - some kind of curve
I cannot understand, or solve, what kind of a curve this inner arc is. Seems like it is a arc, at first look. But at the second, seems like this is a spiral, but I can't prove this...
I would like to get exact equation or parametrization of this curve.
Could you help me with this? Any help is greatly appreciated.
PS: In a first approximation, suppose, that angle of an outer-arc is between 0 and 90 degrees. In the general case this angle can be between 0 and 360 degrees. Lower straight line is always horizontal, 'right' line is always parallel to the radius of an outer-arc.
UPD: Sorry, for misleading, forgot about the main.
This shape is just like a part of a ring with different thickness. We know startThickness and endThickness (lengths of a segments – left straight line and right straight line).
We can choose any angle, cut this 'ring' by a radius at this angle - and the thickness in this cut, should be:
thickness = startThickness + (endThickness - startThickness) * cutAngle/ fullAngle.
So thickness is grown linear from start to end.
So, this image is just like a demonstration, it shows the idea. I don't have task to work with this image.
Seems, like I found part of the solution.
In polar coordinates we can describe arc like this: $$ \left\{ \begin{array}{c} \rho=R \\ \theta=\alpha_1+\alpha_2 t \end{array} \right. ,t \in [\alpha_1, \alpha_1+\alpha_2] $$ where $\alpha_1$ – start angle, $\alpha_2$ – full arc angle
And the bottom curve, (based on the:
$thickness=a+ (b- a)\alpha_2/ \beta$
where $\beta$ is a cut angle, $a$ - start thickness, $b$ - end thickness)
has this system as a parametrization. $$ \left\{ \begin{array}{c} \rho=R-a - (b-a) t \\ \theta=\alpha_1+\alpha_2 t \end{array} \right. ,t \in [\alpha_1, \alpha_1+\alpha_2] $$
But I still don't know what kind of a curve this parametrization describes.
Next step I should translate this system to Cartesian coordinate system, using:
$\rho^2=x^2+y^2$
$\theta=atan2(y, x)$
Reference: Polar coordinate system. Look at 'Converting between polar and Cartesian coordinates' section
UPD: OK, got it, thanks to @achillehui for hint about Archimedean spiral.
From this system: $$ \left\{ \begin{array}{c} \rho=R-a - (b-a) t \\ \theta=\alpha_1+\alpha_2 t \end{array} \right. ,t \in [\alpha_1, \alpha_1+\alpha_2] $$
we can find:
$t=(\theta-\alpha_1)/\alpha_2$
and then:
$\rho(\theta)=R-a - (b-a) (\theta-\alpha_1)/\alpha_2$
UPD 2:
In Cartesian coordinate system:
$x=\rho \cos\theta$
$y=\rho \sin\theta$
So, in this case:
$x(\theta)=(R-a - (b-a) (\theta-\alpha_1)/\alpha_2)\cos\theta$
$y(\theta)=(R-a - (b-a) (\theta-\alpha_1)/\alpha_2)\sin\theta$
$\theta\in[\alpha_1, \alpha_1+\alpha_2]$