function for second quadrant in unit circle

Question

I am no "math-guy" and would really appreciate some help in writing a function for the second quadrant of the unit circle.

Conditions I want to be met:

• Centre of the circle is moved so second quadrant is between 0 – 1 on the x-axis
• Introduce parameters so I can bend the outer edges on the curve (black dashed lines in figure). What I thought of so far is writing two smaller circles if x < value 1 or x > value 2. Value 1 and 2 are values for when the curve should be bent.
• Control the function so that x = 0.5 always returns wanted y-value. (blue dashed line in figure)
• the derivative is equal to 0 at x = 1

Answer 1

You can try the family

$$x^\alpha+y^\alpha=c$$ with $\alpha>2$ and $c\le1$. (Below, $\alpha=3,c=0.6,0.8,1$; the curves should continue to the axis.)

Mirror and translate as required.