Signed volume

As explained in Geometry for Programmers

Signed volume is an oriented measure of the volume spanned by vectors, with sign determined by their ordering and orientation.

Signed volume is the oriented volume of the parallelepiped spanned by three vectors. Compute it as the triple product a · (b × c), or as the determinant of a matrix formed from the vectors. section 9.4.1 section 9.8

Use these rules to interpret the value:

• Read the magnitude as the parallelepiped’s volume and the sign as the vectors’ relative orientation. section 9.4.1
• Rotate the operands cyclically without changing the result: (a × b) · c = (c × a) · b = (b × c) · a. section 9.4.4
• Swap two operands to reverse the sign, and compute the cross product before the dot product. section 9.4.1 section 9.4.4
• In two dimensions, the analogous signed volume becomes the signed area of a parallelogram. section 9.5.2

Distance from a point to a plane

For plane points p1, p2, and p3 and another point p, form (p2 − p1) × (p3 − p1). Its dot product with (p − p1) gives signed volume; dividing by the cross product’s length divides by the base area and yields the point-to-plane distance. Reusing the cross product keeps the computation light. section 9.4.3

Figure 9.20 A parallelepiped formed by p₂-p₁, p₃-p₂, and p-p₁, and the parallelogram formed by p₂-p₁, p₃-p₁, too

Visualize the parallelepiped built from two vectors in a plane and a third vector reaching toward the point; its height represents the distance to the plane. section 9.4.3

How Geometry for Programmers covers it

How deep it goes

Explains geometry, operand ordering, computation, determinants, and a distance application while deferring general high-dimensional determinant algorithms.

Examples

Uses coordinate basis vectors, parallelepipeds, two-dimensional parallelograms, and point-to-plane distance as concrete examples.

What it assumes

Assumes vectors, dot products, cross products, parallelograms, and determinants, while relating their geometric meanings.

What you can do after

Apply it via cross and dot products or determinants to calculate oriented volume and derive point-to-plane distance.

How it connects

Connects signed volume with the triple product, cross product, dot product, determinants, parallelepipeds, and plane distance.

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Related concepts

• triple productproduced by · The triple product is the scalar obtained by taking the dot product of one vector with the cross product of two other vectors, equal to the signed volume of their parallelepiped.
• parallelepipedmeasures · A parallelepiped is a three-dimensional polyhedron with six parallelogram faces arranged in three pairs of parallel planes.
• signed areaequivalent to · Signed area is an oriented two-dimensional area whose sign records the ordering and orientation of the spanning vectors.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.