Finding the intersection envelope
We want to find the formula for the intersection between any two adjacent lines in our bridge.
Let’s look at a line that intersects the axis at a certain point Its function, is 0 at and at 0.
The general equation for such a line is
Now let’s look at a line which intersects the axis at some point and the line just after it, which intersects the axis at a point we call
The curve we are after is the one we’d get if there were infinitely many chords. In other words, as the spacing of our chords on the axis, approaches 0, the point in which our two lines intersect approaches a point on our envelope.
Let’s first find the intersection point of our two lines by setting
As we get
Putting this in we get
So we now know that our curve is defined as all the points that satisfy
for all in [0,1]. So the curve we are after is