# What is this shape?

$C = \{(c_1,c_2):c_1^2 + c_2^2 \leq 1 \}$

$G = \{(g_1,g_2): g_1 = a_1 + d_1, g_2 = a_2 + d_2, d_1^2 + d_2^2 \leq 1 \}$

C is a unit circle centered at the origin, and G is a unit circle centered at $(a_1, a_2)$.

Define:

$X = \{(x_1,x_2): x_1 = c_1 g_1, x_2 = c_2 g_2, (c_1,c_2)\in C, (g_1, g_2)\in G\}$

What is the shape of $X$? Is there any name of it, or any other method to express like a polynomial equation? I thought it might be a ellipse or a circle, but it seems not.

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## 1 Answer

For any point $(g_1,g_2) \in G$, $\{(c_1 g_1, c_2 g_2): (c_1, c_2) \in C\}$ is an ellipse centred at the origin with semi-axes $|g_1|$ (in the $x$ direction) and $|g_2|$ (in the $y$ direction), and thus the equation $(x/g_1)^2 + (y/g_2)^2 \le 1$. Taking $g_1 = a_1 + \cos \theta$ and $g_2 = a_2 + \sin \theta$, the envelope of these ellipses will be determined by the equations $F(x,y,\theta) = 0$ and $F_\theta(x,y,\theta) = 0$, where $F(x,y,\theta) = (\frac{x}{a_1 + \cos \theta})^2 + (\frac{y}{a_2 + \sin \theta})^2 - 1$. In principle you can write $\cos(\theta) = c$ and $\sin(\theta) = s$, eliminate $s$ and $c$ from these equations plus $c^2 + s^2 = 1$, and be left with a polynomial equation in $x$ and $y$. It's looking pretty complicated, though.

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