Spatial geometry 5

Cross Product

For a pair of vectors $u = [x_u, y_u, z_u]$ and $v = [x_v, y_v, z_v]$ the cross product is a vector

$w = u \times v$

where $u \times v = [y_u * z_v – z_u * y_v, z_u * x_v – x_u * z_v, x_u * y_v – y_u * x_v]$

There is absolutely no reason to memorize such a convoluted expression! The key point to appreciate is its geometric meaning.

24KB

3A_3_Cross_product.gh

Orthogonality

The result of this operation is a vector perpendicular to both inputs. So given three points in space, “A”, “B” and “C”, we can define the vectors $u = B - A$ and $v = C - A$ and compute $n = u \times v$. This vector is called the “normal” vector i.e. perpendicular to the plane spanned by $u$ and $v$.

Practice

Tasks

• Use the dot product to verify that the cross product vector is perpendicular to both input vectors.

• What is the result of one of the input vectors being the zero vector to the cross product vector?

Hints

Parallelism

If the input vectors are parallel with one another then the cross product is the zero vector. Mathematically parallelism is equivalent of one vector being a multiple of another in the scaling sense i.e. $u = s * v$. The result of a cross product being zero expresses the idea of infinite choices for selecting a perpendicular vector to the supplied input rather than the impossibility to finding one.

Practice

Tasks

• Construct two dependent vectors and verify using the vector length that their cross product is zero.

• Explore the trigonometric form [>] of the cross product and consider the effect of different values of the angle between the input vectors.

Hints

Directionality

Unlike other arithmetic operations, the order of multiplication is important: reversing the order, reverses the direction of the result i.e.

$n = u \times v$

$v \times u = -n$

We use the right or left hand fingers, with the index being “u”, the middle “v” and the thumb as “n” vectors to capture this concept. In the video below, notice how the normal vector flips direction when the point order viewed from the top switches from clockwise to counter-clockwise.

Practice

• Construct a triangle and point floating about in the same plane. Compute all pairwise cross products between the triangles edges and vectors from each point to the floating point. What can we infer from the directions of the cross products about the point’s relationship to the triangle?

• Construct a polygon and apply the cross product for each consecutive edge. Explain what we can conclude about their points by comparing shapes like a quadrilateral versus a dart or bow tie shape considering the direction of the point-wise cross products.

Computing Area

The magnitude of the cross product encodes the area of the parallelogram spanned by the two vectors or twice the size of the triangle spanned. The expression for area of triangle is thus concisely capture by $|u \times v| / 2$ instead of the $base * height / 2$ which implies decomposition of the original triangle into an orthogonal triangle and projection just to measure its height etc.

Area of parallelogram

Practice

Tasks

• Construct a closed polygon with five points i.e. an irregular pentagon and compute its area using the cross product.

• Can the area of four-point bow tie shaped polygon be computed using the cross product rule? How about a dart shaped quadrilateral?

Hints

Projection

In as much the dot product “drops” dimensions i.e. expresses projection in the sense of perpendicular-to, the cross product “raises” a dimension i.e. expresses a notion of "unprojection" or lifting a perpendicular-from.

Practice

Tasks

• Given three points of an equilateral triangle, compute the centroid and raise a point such that it forms the fourth point of a regular tetrahedron [>].
• Construct a square and lift two points from its centre such that they describe a regular octahedron [>].

Hints

Orthonormalization

For geometric constructions it is very useful to work with orthogonal and unit length vectors triplets. This is because translations in each direction does not interfere with one another and defining the amount of travel is as simple as scaling the unit basis vectors. The process of constructing such vectors is known as orthonormalization.

Given two arbitrary non-parallel vectors $u$ and $w$ we can construct a normal $n$ square to both using the cross product. By repeated application of cross product between $u$ and $n$ we can construct vector $v$ such that now all $u$, $v$ and $w$ are orthogonal to one another. We typically normalize the vectors to form what is known as spatial basis $[\hat{u}, \hat{v}, \hat{w}]$ or coordinate system or frame $[o, \hat{u}, \hat{v}, \hat{w}]$.

Practice

Tasks

• Use the “plane from 3 points” component and observe its axes. It replicates the above demonstrated orthonormalization process.

• Use the “construct plane” component and supply two non-orthogonal vectors. Verify that the plane’s axes are automatically orthonormalized by using the “deconstruct plane” component.

Hints