圆的变换

Transforming A Circle

Translation

The Formula

We start with the general equation of the circle, $x^2 + y^2 = r^2$. To translate this to the right $h$ units, we change the term $x^2$ to $(x - h)^2$. Similarly, to translate the circle up $k$ units, the $y^2$ term becomes $(y - k)^2$. Thus our formula to translate a circle to the right $h$ units and up $k$ units is: $(x - h)^2 + (y - k)^2 = r^2$

Why?

Intuitively, one would assume that to move a circle right $h$ units and up $k$ units, we would add $h$ to $x$ and add $k$ to $y$. However, preforming these additions moves the circle to the left $h$ units and down $k$ units. The concept is the same as translating a function. As we subtract $h$ from $x$, it takes another $h$ movements to the right to obtain that same y-coordinate as we had before. For example, given a circle with equation $x^2 + y^2 = 36$. When we translate the circle to the right 3 units, given our new formula, our new equation is $(x-3)^2 + y^2 = 36$. Using x-coordinates 0 to 6, we can create a table of values for each equation as we solve for y.

$$x^2 + y^2 = 36$$ $$\begin{array}{r|l}x & y \\\hline 0 & 6 \\ 1 & \sqrt{35} \\ 2 & 4\sqrt{2} \\ 3 & 3\sqrt{3} \\ 4 & 2\sqrt{5} \\ 5 & \sqrt{11} \\ 6 & 0\end{array}$$ $$(x - 3)^2 + y^2 = 36$$ $$\begin{array}{r|l}x & y \\\hline 0 & 3\sqrt{3} \\ 1 & 4\sqrt{2} \\ 2 & \sqrt{35} \\ 3 & 6 \\ 4 & \sqrt{35} \\ 5 & 4\sqrt{2} \\ 6 & 3\sqrt{3}\end{array}$$

Do you see the correspondence? Three units to the right, the y-coordinates for the new equation were the same as the old equation. X-coordinates were a little confusing from 0-2 because the point $(3,6)$ is the top point in the top half of the curve. Translating a circle up or down is the same concept. I hope this clears any confusion for translating a circle.

Rotation and Reflection

Since a circle consists of every point a certain distance away from the center of the circle, reflecting or rotating a circle would have no effect.

Dilation To dilate a circle, we start with our standard equation: $x^2 + y^2 = r^2$ To dilate the circle we multiply our desired factor squared into the right side of the equation. For example, two multiply the diameter of the circle by two, our equation would now be $x^2 + y^2 = 2^2(r^2)$.