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How does one prove the "radius" of a circle is invariant of its location, and in...
Jan 29, 2024
How does one prove the "radius" of a circle is invariant of its location, and in such a way that this can be applied to other distance-based shapes? Analysis I'm trying to understand how this might work for ANY shaped defined by a Jordan curve, which remains translation invariant in a certain sense. Let's say you have a circle as a map from R∧2→RR^{\wedge} 2 \rightarrow R marked by \operatorname{sqrt}\left(x^{\wedge} 2+y^{\wedge} 2\right)<=r. How do I prove that it doesn't matter where the circle is, the radius rr is still the same? If I have sqrt (x−a)∧2+(x−b)∧2)\left.(x-a)^{\wedge} 2+(x-b)^{\wedge} 2\right), it's a very different formula, and yet intuitively, you can re-draw the same circle on a piece of paper and it retains its radius. How is this shown mathematically?
Solution by Steps
step 1
Consider a circle with center at the origin (0,0) and radius r r in the Cartesian plane. The equation of this circle is x2+y2=r2 x^2 + y^2 = r^2
step 2
Translate the circle by a vector (a,b) (a, b) , so the new center is at (a,b) (a, b) . The equation of the translated circle is (x−a)2+(y−b)2=r2 (x - a)^2 + (y - b)^2 = r^2
step 3
Observe that the radius r r does not change in the equation after translation. The terms (x−a)2 (x - a)^2 and (y−b)2 (y - b)^2 represent the squared distances from the new center (a,b) (a, b) to a point (x,y) (x, y) on the circle
step 4
The invariance of the radius r r under translation is due to the fact that translation is a rigid motion, which does not alter distances within a geometric figure
Answer
The radius r r of a circle is invariant under translation.
Key Concept
Invariance of radius under translation
Explanation
The radius of a circle remains unchanged when the circle is translated because translation is a rigid motion that preserves distances within a shape. The equation of the circle changes to reflect the new center, but the radius r r in the equation remains the same, indicating that the radius is invariant under translation.
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