Linear homotopy

Definition

Suppose $U$ is a subset of $\mathbb {R} ^{n}$ and $f,g:X\to U$ are continuous maps. Suppose further that for every $x<\in X$, the line segment joining $f(x)$ to $g(x)$ lies completely inside $U$. The linear homotopy between $f$ and $g$ is a map as follows:

$x\mapsto (1-t)f(x)+tg(x)$

where the computation on the right side is in $\mathbb {R} ^{n}$. Essentially we are moving from $f(x)$ to $g(x)$ along a straight line.

A composite of several linear homotopies is termed a piecewise linear homotopy. If there exists a piecewise linear homotopy between two functions $f,g:X\to U$ then we say that $f$ and $g$ are piecewise linearly homotopic maps.