What is meant by a "signed" area? I don't see any circumstance in this situation where we might possibly want to define a negative area.

@jchh I did some googling and I think one reason why it has to be a signed area is the case where the point x is outside of the triangle. In order for the area of the resulting sub-triangles to still add up to the area of the main triangle, some of the areas need to be negative. To better visualize it, think about x being super far from the triangle and draw those white lines. You're resulting sub-triangles are gonna be pretty big compared to the original triangle so without negative areas things won't add up properly. (And I think the signed areas of the sub-triangles adding up to the area of the main triangle is the same thing as alpha, beta, and gamma adding up to 1). Here's where I read about it (page 4). There's a helpful picture. https://www.geometryexplorer.xyz/pdfs/Barycentric%20Coordinates.pdf

Why would someone prefer one way of calculating barycentric coordinates as opposed to the other? Are there any significant differences?