Homothetic function versus straight Engle curve
A general result using indirect utility functions, is that expenditure linear in income ("straight-line" Engel curves) is produced if and only if the indirect utility function has the "Gorman-polar form".

The original paper is

Gorman, W. M. (1961). On a class of preference fields. Metroeconomica, 13(2), 53-56.

...where we read that a linear Engel curve is produced when we have a homothetic utility function (Gorman does not use the word, he writes "homogeneous" and in a footnote, enhances that to "function of a homogeneous function", which is how a homothetic function is defined).

But the reverse does not hold:homotheticity is sufficient for linear Engel curves, but not necessary.

For example, linear Engel curves are also produced by a utility function that reflects "quasi-linear" preferences, $U(x,m) = u(x) + m$, which are not homothetic.



A homothetic function is properly defined mathematically as follows:

Let $f(\mathbf x)$ be a homogeneous function of some degree, and let $g$ be a function with non zero derivative. Then $g[f(\mathbf x)]$ is called a homothetic function. In economics, we usually require for the derivative of $g$ to be strictly positive, so that we remain inside economically meaningful territory.