I was trying to work out what path a ladder traces as it slides down a wall without slipping. There is plenty of info out there about the loci of single points on the ladder, e.g. the midpoint traces a circle, any other point traces an ellipse. However, I am concerned with the locus of the entire ladder.
The ladder AB, which is at an angle $\theta$ to the ground and is of arbitrary length 5, has a point $n$ units along its length. Considering the point where the wall meets the ground as the origin, using trig we can see that $x = (5-n)cos(\theta)$ and $y = nsin(\theta)$. This is effectively a parametric equation with parameter $\theta$ and restriction $0 < \theta < \pi/2$. Graphing the locus of this curve for values of $n$ from 0 to 5 gives the following (ignore the other 3 quadrants - I wan't sure how to limit the values of the parameter in GeoGebra):
The circle graphed there for comparison shows the curve isn't quite circular. So, my question is: what is this curve? Can it be described by an equation?
The end of the ladder against the wall traces a vertically downwards path (scraping against the wall) and the end of the ladder resting on the ground traces a horizontally sideways path (scraping along the ground).
Any point in between the two ends would be a combination of vertical & horizontal movements. At no point during the slide of the ladder along the wall, can any part of the ladder rise vertically i.e. I don't see how a circular/closed loop path can occur.