1. What is the equation describing this pseudo cardioid's shape?
2. What is the length of the curve?
3. What is the area of the curve?
4. Does it bear some simple relation to an actual cardioid?
5. Is this a new curve or does it have a name?

Let 2a be the angle CAB, and let t be chosen so that sin a = 2t/(t^2 + 1) and cos a = (t^2 - 1)/(t^2 + 1)

It follows that cos 2a = (t^4 - 6t^2 + 1)/(t^2 + 1)^2
and sin 2a = 4t(t^2 - 1)/(t^2 + 1)^2

Therefore the location of the centre of the incircle is at:

((t - 1)^2/(t^2 + 1), 2t(t - 1)/(t^2 + 1)(t + 1))

and naturally the radius is equal to the y co-ordinate.

Therefore the equation of the incircle is:

x^2 + y^2 - 2*x*(t - 1)^2/(t^2 + 1) - 4*y*t*(t - 1)/((t^2 + 1)*(t + 1)) + ((t - 1)^2/(t^2 + 1))^2 = 0

For the points on the envelope, the derivative of this expression with respect to t must also be 0. This gives us two equations in the variables x and y. Solving with maple and neglecting the solution on the x axis gives:

x = (t^10-t^8-12*t^7+8*t^6+8*t^5+8*t^4-12*t^3-t^2+1)/((t^2+1)*(t^8+2*t^6+2*t^4+2*t^2+1))

y = 2*t*(t^8+2*t^7-2*t^6-6*t^5+6*t^3+2*t^2-2*t-1)/(t^10+3*t^8+4*t^6+4*t^4+3*t^2+1)

since I could see no suitable substitution to simplify this, I will leave the work of calculating the length, area, and so on to those with more guts.

My best guess is that this curve does not have a name.

Thankyou

sfwc
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I don't really know what you mean by an 'envelope of circles'. I'm used to an 'envelope' being a series of straight lines each tangent to a curve.

If you mean the locus of the centres of the circles, the equation I get is

r = sec(a/2)*tan(pi/4-a/4) / (tan(a/2) plus tan(pi/4-a/4))

but I don't care to integrate it. :-)

And it doesn't look particularly like a cardioid.