I’ve been digging through (and talking about) definitions of different terrain attributes this week, and one question I got a few times was, “What does orthogonal mean?” Every time I tried to define orthogonal, I used the word perpendicular, so it seemed wise to just define that one today too.
Let’s start with perpendicular. This is a word you probably learned about in elementary geometry. Perpendicular describes two objects, usually lines or line segments, that meet at a 90 degree angle. The adjacent sides of a square are perpendicular. Two perpendicular lines intersect once and are symmetric.
The tricky thing about orthogonal and normal is that they mean the same thing as perpendicular, but are used in different, specific contexts. Orthogonal is a word you might have come across in college math or linear algebra, and it’s usually used in the context of multidimensional space or with matrix math. Another place you may have heard the word orthogonal is in relation to running contrasts after an ANOVA. To run contrasts, you make an orthogonal matrix to make sure that the contrasts you’re running are independent (learn more about that process here)
Normal is another word you’ll hear thrown around with perpendicular and orthogonal. In geometry context, normal usually has to do with planes that meet at a right angle whereas perpendicular usually refers to lines. In matrix math, normal usually means that a vector is both orthogonal and of unit length (this can also be called ortho-normal).
The important thing to remember is that all these words have to do with right angles. Sometimes orthogonal has to do with independence, and sometimes normal has to do with unit length. If you aren’t sure if that’s the case in a certain context, you can usually check Stack Exchange for a similar example or ask someone. There are lots of resources out there!