How to make an angle smooth

Question

Suppose we are given in $R^2$ an angle ACB and a circle centered at C, like the following picture shows.

How can we "draw" a $C^1$ curve containing the segments outside of the circle and the part AB would be inside the circle and angle at the same time?

Answer 1

You can easily solve for a quadratic polynomial that will make this $C^1$ (or, indeed, a quartic polynomial that will make it $C^2$). Just take $y=f(x)=ax^2+bx+c$, set $f(\pm x_0)=0$ and $f'(\pm x_0) = \mp m$. You have to specify $x_0$ and $m$ appropriately to make sure the curve stays below $C$. (Here I'm taking the $x$-coordinate of $C$ to be $0$ and the $x$-coordinates of $A$ and $B$ to be $\pm x_0$ respectively.)

Answer 2

You already have slopes and values of the desired curve at A and B. Pick a point P inside the circle and below C.

Find a fourth degree polynomials with the given information and you are done.

Answer 3

I present to you: the Bézier Curve.

We parameterize the curve with a time value $t$ that varies from $0$ to $1$. Then, to find a point on the curve, we generate points: $$D_t = (1-t)A+tC$$ $$E_t = (1-t)C+tB$$ $$F_t = (1-t)D_t+tE_t$$

The locus of $F_t$ as we vary $t$ is then our curve.

Here's one Bézier curve, drawn with 16 segments.

If you want more control, you can switch to rational Bézier curves, which work approximately the same way, but include a weight with each control point. Then we have, with weights $a$, $b$, and $c$ corresponding to the three control points above: $d_t = (1-t)a + tc$, etc, mirroring the control equations from before. Then our final points on the curve are $$F^*_t = \frac{F_t}{f_t}$$ This is equivalent to placing our control points in homogeneous coordinates with varying $w$ values. In all cases with non-zero weights, the resulting curves are tangent to the lines between the control points at the ends. There is a weight value that makes the curve a circular arc, equal to the sine of half the angle at the center.

Here's a series of rational Bézier curves, with endpoint weights $1$ and center control point weight varying from $0.1$ to $10$.