I recently read that a line can be defined using the formula $$ A = O + dL $$ where $A$ represents any point on the line, $O$ represents the vector origin of the line, $L$ represents the direction vector of the line and $d$ represents the distance along said line from the origin.
This formula makes it possible to define a line as an infinite series of points, dependent on the variable d.
My question is; can the same be done with a rectangular prism? A prism is a three-dimensional object, so I would expect three variables to be part of the equation rather than one, however I have no idea how to go about defining it using a formula. More specifically, I would like to be able to define the volume inside the rectangular prism as an infinite series of points (much like the line formula), such that the formula takes the form $$ R = F(w, d, l) $$ where $R$ is a vector describing any point within the volume and $w$, $d$ and $l$ are variables ranging from $0$ to $1$, denoting the distance along the width, depth and length of the rectangular prism respectively.
Thanks in advance :)
The answer is an resounding no! What you have read is true for affine subspaces, and this dl is a linear operator acting on l, so you need a two-dimensional vector space, and prism is not. So, if A and C are the most far-away points of prism, those ones exist, because prism is compact, so every function, and in our case it is distance function, has its max on the compact set. Because it contains a vector joining A and C, it also must contain a line between them, or otherwise, it must contain tAC for every t, say t = 2, from A moving with 2AC we get something that is not in a prism and that is a contradiction.