# What is the difference between "linear" and "piecewise linear" and "exponential way"?

When the impact of laws diminish over time, it may dissipate by "linear", "piecewise linear" or "exponential way".

It is easy to understand that linear mean that the the erosion of the laws impact follows a constant slope coefficient. And exponential way is that the erosion of laws effect would follow a curve.

But I do not fully understand what does "piecewise linear" mean intuitively? Is it the mixture between "linear" and "exponential" ? Could you please explain it to me, thanks in advance.

• What about breaks in linear trends due to some events? Or it could be just some mathematical model for approximation. Sep 30 at 14:50
• @chan1142 , I am just simply talking about some mathematical model Sep 30 at 19:36

I think a picture would best convey the intuition. The red line in the graph below shows a linear trend, the blue piece-wise linear, and the green exponential. The PWL (piece-wise linear) model replaces a curve with a sequence of straight line segments joined together to approximate the curve.

I am not familiar with the use of PWL in economics, however, it is the same principle used in science and engineering. This two page reference shows a PWL for a model rocket motor:

https://www.nar.org/SandT/pdf/Estes/A10-0T.pdf The data for a typical thrust-time curve are measured with short sampling time and therefore high sampling rate. This data sampling model is a PWL model with many dense line segments joined to each other to make the curve.

The PWL model below approximates the PWL curve above using fewer line segments for use with a standard rocket simulation software package with a limit of 32 data points for the PWL motor model: 