Function to express a stick sliding until it hits the floor

Question

I have a stick standing leaning against a wall.

On the exact middle of the stick, I have painted a red dot.

The stick is sliding until it hits the floor.

What figure does the red dot 'draw in the air' and how do I find the function expression for this 'pattern'/'figure'?

I have painted this figure of the situation:

The black line is the wall and the grey line is the stick with the red dot. The stick is first standing against the wall (figure 1) and then slides slowly until it hits the floor (figure 4).

I guess it follows one of these patterns (green path, blue path, pink path), and it might be possible to draw this pattern using a trigonometric function.

Answer 1

Let $\alpha$ be the angle between stick and floor. Let point $(0,0)$ (the beginning of the coordinate system) be in the meeting between floor and wall. Suppose that sitck has length $1$. Left end of stick (point $A$) has coordinates: $(- cos \alpha, sin \alpha) $ and the red point is translated from point A about vector $[\frac{1}{2} cos\alpha, \frac{1}{2} sin \alpha] $. Hence the red point has coordinates: $(-\frac{1}{2} cos\alpha, \frac{1}{2} sin \alpha)$ and this is the circle (green trajectory).

Answer 2

Lets say that they foot of the stick is $x$ units from the wall. The top of the stick is $y$ units from the floor.

the midpoint is $(x',y') = (\frac x2, \frac y2)$

And the length of the stick is fixed $\sqrt {x^2 + y^2} = L\\ x^2+y^2 = L^2$

What equation expresses $(x',y')?$

$x'^2 + y'^2 = {(\frac L2)}^2$

Answer 3

The blue segment has constant length.

Answer 4

I find arkeet's answer very good. Alternative point of view:

At every moment of the stick movement let's imagine the stick mirrored by a horizontal plane fixed to (i.e. moving with) the red middle point. This way the image middle point is always at the stick red point. The image is a stick that revolves around its lower end. Every point of the image (besides the said end) draws a quarter of its own circle; also does the image middle point which is the stick red point.