Exercises

These are exercises accompanying the book Projective Geometric Algebra Illuminated. More will be posted over time.

Exercises for Chapter 2

1. Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$.

2. Let $$\mathbf u$$ be a basis element of the 4D projective algebra. Prove that if $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ and $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$, then it must also be true that $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$ and $$\underline{\mathbf u} \vee \mathbf u = \mathbf 1$$. That is, show that right and left complements under the wedge product are also the right and left complements under the antiwedge product.

3. Suppose that the 4D trivectors $$\mathbf g$$ and $$\mathbf h$$ represent parallel planes in 3D space. Show that the magnitude of the moment of $$\mathbf g \vee \mathbf h$$ is the distance between the planes multiplied by both their weights.

4. Let $$\mathbf m$$ be a $$4 \times 4$$ matrix that performs a rotation about the $$z$$ axis in homogeneous coordinates. Calculate the $$16 \times 16$$ exomorphism matrix $$\mathbf M$$ corresponding to $$\mathbf m$$.

5. Suppose that $$\mathbf G$$ is a metric exomorphism. Use the fact that $$\mathbf G$$ is an exomorphism to prove that the associated antimetric $$\mathbb G$$ must satisfy $$\mathbb G(\mathbf a \vee \mathbf b) = \mathbb G\mathbf a \vee \mathbb G\mathbf b$$ for any $$\mathbf a$$ and $$\mathbf b$$.

6. Suppose that the metric tensor $$\mathfrak g$$ is invertible. Show that the wedge and antiwedge products satisfy the relationship $$\mathbf a \vee \mathbf b = (\mathbf a^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605})^\unicode["segoe ui symbol"]{x2606}$$.

7. Suppose that $$\mathbf a$$ and $$\mathbf b$$ are basis elements of an $$n$$-dimensional exterior algebra and $$\operatorname{gr}(\mathbf a) + \operatorname{gr}(\mathbf b) = n$$. Show that $$(\mathbf a \wedge \mathbf b)^\unicode["segoe ui symbol"]{x2605} = \mathbf a^\unicode["segoe ui symbol"]{x2605} \mathbin{\unicode{x25CF}} \mathbf b$$.

8. Show that the geometric norm is idempotent. That is, show that $$\Vert \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}} \Vert = \mathbf a\mathbf 1 + \mathbf b{\large\unicode{x1D7D9}}$$.

9. Derive the relationship between left and right interior products shown in Equation (2.110).

10. Derive Equation (2.159), which is the expansion analog of Equation (2.129).

11. Derive a formula for $$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}}$$, the double bulk dual of $$\mathbf u$$, that uses only $$\operatorname{gr}(\mathbf u)$$, $$\operatorname{ag}(\mathbf u)$$, and the determinant of the metric tensor $$\mathfrak g$$.

12. Assuming that the antivector basis elements are written in the same order as their vector complements, prove that the $$(n - 1)$$-th compound matrix $$C_{n - 1}(\mathbf m)$$ of a matrix $$\mathbf m$$ is always equal to the adjugate transpose of $$\mathbf m$$.

13. Develop the two-dimensional projective exterior algebra having the basis elements $$\mathbf 1$$, $$\mathbf e_1$$, $$\mathbf e_2$$, and $${\large\unicode{x1D7D9}} = \mathbf e_{12}$$. Give the metric exomorphism for this algebra, give the homogeneous representation of a one-dimensional point and identify its bulk and weight, give the bulk dual and weight dual of a point, and give the geometric norm of a point.

Exercises for Chapter 3

1. Determine in what numbers of dimensions the reverse and antireverse operations are equivalent, each always producing the same results as the other.

2. Derive a translation operator, not necessarily unitized, that moves a given 3D point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ to the origin.

3. Show that the geometric constraint is closed under the geometric product. That is, if $$\mathbf a$$ and $$\mathbf b$$ both satisfy the geometric constraint, show that $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ also satisfies the geometric constraint.

4. Derive a formula that gives the square root of the complement motor $$\mathbf Q$$ appearing in Equation (3.221).

5. Formulate motors and flectors in the two-dimensional projective exterior algebra that models one-dimensional space. What geometric transformations does each type of operator perform?

Exercises for Chapter 4

1. In the 5D conformal geometric algebra modeling 3D space, show that $$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}} = -\mathbf u$$.

2. Calculate the wedge product of two round points having the same center but different radii, and interpret the result.

3. Let $$\mathbf s_1$$ and $$\mathbf s_2$$ be two intersecting unitized spheres that have real radii. Show that the dot product $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2$$ is equal to the product of the radii multiplied by the cosine of the exterior angle wherever the surfaces of the two spheres meet.

4. Consider the 3D conformal geometric algebra modeling 1D space, where $$\mathbf e_2$$ represents the origin and $$\mathbf e_3$$ represents the point at infinity. Determine formulas for the join of two round points $$\mathbf a$$ and $$\mathbf b$$, the meet of two dipoles $$\mathbf d$$ and $$\mathbf f$$, the meet of a dipole $$\mathbf d$$ and a flat point $$\mathbf p$$, the weight expansion of a round point $$\mathbf a$$ onto a dipole $$\mathbf d$$, and the weight expansion of a round point $$\mathbf a$$ onto a flat point $$\mathbf p$$.

See Also

• PGA Illuminated on Amazon.com
• Contents