Mathematics



A new way to draw graphs of functions.

When a function is plotted in an ordinary coordinate system,
the curvature is not the second derivate.
As a civil engineer, I have wondered how to plot a graph so the curvature og the graph is the second derivate.
The program can do so.

The x-coordinate is along the curve.
The derivative in a point is the angel of the curve.
The second derivative is the curvature of the curve.

Drawing the graph is done point by point.
First point is the startpoint.
The derivate in that point is the angle.
A line is drawn with the length, dx, in the angle, f'(x).
This makes a new point.
The derivate in that point is the angle of the curve in that point relative to the curve.
A new line is drawn from this point with the with the length, dx, and the angle, f'(x), relative to the curve.
So continues from the start point to the end point.

The x-axis is not a fixed line but along the curve.

The second derivate is the curvature of the graph.

The graphs can be drawn on a computer.


These graphs have the following properties.

f(x) = x makes a straight line.
f'(x) is constant so the slope doesn't change.

f(x) = x2 makes a circle.
f'(x) changes with x so the slope changes with x.
f''(x) is constant so the curvature is constant.
For example f(x) = x2 / 2 has the curvature 1 and makes an unit circle.

sin(x) and cos(x) become equal to the graphs in the ordinary coordinate system.

e(x) becomes a spiral that turns into itself.
The angle, e'(x), and the curvature, e''(x), etc are accelerating.
e'(x) becomes equal to e(x), e''(x) becomes equal to e'(x) and e(x) and so on.


A program, written in Java, which can plot graphs this way, can be downloaded here.