Chapter 1: Euclidean Geometry

1.1 The Dot Product

The dot product is the foundational operation of Euclidean geometry, encoding both length and angle through a single algebraic construct. Everything in this chapter flows from its definition.

These three properties — bilinearity, symmetry, and positive definiteness — together make the dot product an inner product. It is positive definiteness in particular that distinguishes Euclidean geometry from the Minkowski geometry we will encounter in the hyperboloid model later.

Now that we have an inner product, we can define length in terms of it.

The Polarization Identities will reappear in a crucial way: they tell us that the dot product is entirely determined by the norm, which will be the key step in classifying isometries algebraically. With length in hand, we can define distance.

These three properties define a metric space. A key goal of this course is to construct analogous distance functions on the sphere and the hyperbolic plane, and to verify that they too satisfy these axioms. With distance established, we can now turn to angle, the other fundamental measurement in geometry.

Note that Cauchy-Schwarz guarantees the right-hand side lies in \([-1,1]\), making the definition well-formed.

1.2 Orthogonal Projections

Orthogonal projections let us decompose any vector into components parallel and perpendicular to a given subspace. This decomposition is central to everything that follows, including the classification of isometries.

The proof uses the standard technique of showing \(U \cap U^{\perp} = \{\mathbf{0}\}\) (if \(\mathbf{x} \in U \cap U^{\perp}\) then \(\mathbf{x}\cdot\mathbf{x} = 0\) so \(\mathbf{x} = \mathbf{0}\)) and dimension counting.

The direct sum decomposition \(\mathbb{R}^n = U \oplus U^{\perp}\) is the engine behind orthogonal projection. Every vector has a unique splitting into a component inside \(U\) and a component perpendicular to it.

The following theorem gives the geometric interpretation: the projection is literally the closest point.

When we have an explicit basis for \(U\), there is a matrix formula for computing the projection.

When the columns of \(A\) form an orthogonal or orthonormal basis, the formula simplifies significantly.

1.3 The Cross Product

The cross product is specific to \(\mathbb{R}^3\) and provides a way to produce a vector orthogonal to two given vectors, encoding both orientation and area. It will be essential for computing angles on the sphere and for working with great circles.

The triple product \(\det(\mathbf{u},\mathbf{v},\mathbf{w})\) gives the signed volume of the parallelotope spanned by \(\mathbf{u},\mathbf{v},\mathbf{w}\).

The sign of the triple product detects orientation, leading to the following definition.

1.4 Geometry in the Euclidean Plane

In the Euclidean plane \(\mathbb{R}^2\), angle, line, and circle theory come together into classical geometry.

Triangle Geometry

With the tools of angle and distance established, we can now develop the classical theory of triangles. The central result is one that will break down in both spherical and hyperbolic geometry.

This is a fundamental property distinguishing Euclidean geometry from both spherical and hyperbolic geometry, where the angle sum differs from \(\pi\).

Triangle Centres

Beyond the basic metric properties of triangles, classical Euclidean geometry is rich with special points associated to each triangle — centres defined by different geometric concurrence conditions.

The Euler line is a striking example of how the basic metric structure of a triangle coerces several independently defined points into a single line. The incentre, defined by angle bisectors rather than side bisectors, does not generally lie on the Euler line.

1.5 Isometries of Euclidean Space

An isometry is a distance-preserving bijection. Understanding isometries is the unifying theme of the entire course — each geometry is characterized by its group of isometries.

Orthogonal matrices are precisely those that preserve the dot product: \((A\mathbf{u})\cdot(A\mathbf{v}) = \mathbf{u}^T A^T A \mathbf{v} = \mathbf{u}\cdot\mathbf{v}\). Equivalently, their columns form an orthonormal basis. This connects algebraically to the geometric notion of distance preservation.

The following theorem is the algebraic backbone of the classification: every isometry is rigid, affine, and built from an orthogonal linear part and a translation.

This is where the Polarization Identity pays off: knowing that \(L\) preserves lengths allows us to recover that it preserves dot products, and dot products determine the linear structure completely. The determinant of the orthogonal part \(A\) is \(\pm 1\), giving rise to a natural dichotomy.

Isometries in \(\mathbb{R}^2\)

The isometries of the plane are:

• Identity: \(I(x) = x\).
• Translation: \(T_u(x) = x + u\).
• Rotation about \(p\) by \(\theta\): \(R_{p,\theta}(x) = p + R_\theta(x-p)\) where \(R_\theta = \begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{pmatrix}\).
• Reflection in line \(L\) through \(p\) perpendicular to \(u\): \(F_L(x) = x - \frac{2(x-p)\cdot u}{|u|^2}u\).
• Glide reflection: \(G_{u,L} = T_u \circ F_L = F_L \circ T_u\) when \(L\) is parallel to \(u\).

Translations, rotations preserve orientation; reflections, glide reflections reverse it.

The next theorem reveals that reflections are in some sense the “atoms” of planar isometries — every isometry is a product of reflections. Understanding how reflections compose gives us all the others.

Isometries in \(\mathbb{R}^3\)

In \(\mathbb{R}^3\) the full list expands:

The new entry in dimension three compared to dimension two is the twist (also called a screw motion): a simultaneous rotation about an axis and translation along that same axis. This has no two-dimensional counterpart, illustrating how the classification of isometries is genuinely sensitive to the dimension.

Chapter 2: Spherical Geometry

2.1 The Sphere and Spherical Distance

Spherical geometry lives on the unit sphere \(S^2\), where “straight lines” become great circles and angles sum to more than \(\pi\) in a triangle. This is the simplest model of a non-Euclidean geometry with positive curvature.

Having developed the full toolkit of Euclidean geometry, we now ask: what happens when we replace the flat plane \(\mathbb{R}^2\) with a curved surface? The sphere \(S^2 \subset \mathbb{R}^3\) is the most natural first candidate, and we already have all the tools — dot products, cross products, projections — to work with it.

The spherical distance between two points is simply the angle between the corresponding unit vectors — it measures how far apart they are along the surface of the sphere, not through the ambient space.

This tells us the two distances are monotonically related — one can be recovered from the other — but they are not proportional. Spherical distance is not simply Euclidean distance in disguise.

This is Archimedes’ hat-box theorem: the sphere and the enclosing cylinder have the same lateral area. The elegant cancellation in the integrand — the sphere bows outward exactly as the cap shrinks — is what makes the area formula so clean.

2.2 Spherical Circles and Lines

Note the contrast with Euclidean geometry: the circumference is \(2\pi\sin r \leq 2\pi r\), reflecting the positive curvature of the sphere.

Now that we have a notion of distance on \(S^2\), we can ask: what plays the role of straight lines? In Euclidean geometry, lines are geodesics — shortest-distance curves. On the sphere, the shortest paths between two points run along great circles.

Part (2) is a key departure from Euclidean geometry: there are no parallel lines on the sphere. Every pair of great circles meets, just as every pair of longitudes meets at the poles. The “parallel postulate” fails on \(S^2\), which is what makes spherical geometry non-Euclidean.

2.3 Oriented Angles on \(S^2\)

To measure angles between curves on the sphere, we need to work in the tangent space at each point — the natural home for directional information.

The tangent space \(T_\mathbf{u}\) is simply the plane in \(\mathbb{R}^3\) perpendicular to \(\mathbf{u}\) through the origin — it inherits the full inner product structure from \(\mathbb{R}^3\), which lets us measure angles between tangent vectors using the same formula as in \(\mathbb{R}^2\).

The tangent vector formula is just the normalized version of the component of \(\mathbf{v}\) perpendicular to \(\mathbf{u}\) — geometrically, it is the direction you would initially travel along the great circle from \(\mathbf{u}\) toward \(\mathbf{v}\).

2.4 Spherical Triangles

With a notion of line segments and angles on \(S^2\), we can now define and study triangles. Spherical triangle geometry is both richer and more symmetric than its Euclidean counterpart.

This is one of the most beautiful theorems in geometry: the area of a spherical triangle is completely determined by its angle sum, with no reference to side lengths.

The next construction introduces a remarkable symmetry unique to spherical geometry: the polar triangle, which swaps the roles of sides and angles.

The polar triangle is a duality: sides and angles exchange roles up to supplementation. This allows us to derive the Second Law of Cosines from the First.

The Second Law is derived by applying the First Law to the polar triangle using the duality \(a \leftrightarrow \pi - \alpha'\), etc.

The AAA congruence corollary captures one of the most striking differences between spherical and Euclidean geometry: there is no concept of similar-but-not-congruent triangles on the sphere. Two spherical triangles with the same angles are necessarily the same size.

2.5 Isometries on \(S^2\)

Just as isometries of \(\mathbb{R}^n\) were characterized as maps of the form \(A\mathbf{x} + \mathbf{b}\), isometries of \(S^2\) have a clean algebraic description — and the ambient \(\mathbb{R}^3\) makes it transparent.

This mirrors the Euclidean Theorem 1.83: the composition law for reflections has the same form on \(S^2\) as in \(\mathbb{R}^2\). This is not a coincidence — it reflects the local flatness of the sphere (at any point, the geometry looks Euclidean to first order).

2.6 Projections of \(S^2\)

Several important projections map parts of \(S^2\) to flat regions. Each has distinctive properties.

No single projection can faithfully represent all properties of the sphere on a flat map — there is always some distortion. The choice of projection depends on which property one wishes to preserve: area, angle, or straightness of geodesics.

Orthogonal Projection: The map \(\phi: H \to D\) from the upper hemisphere \(H\) to the unit disc \(D\) given by \(\phi(x,y,z) = (x,y)\). The inverse is \(\psi(u,v) = (u, v, \sqrt{1-u^2-v^2})\).

Lambert Cylindrical Equal-Area Projection: The map \(\phi: S \to R\) (where \(S = S^2 \setminus \{\pm(0,0,1)\}\) and \(R\) is the rectangle \([0,2\pi)\times[-1,1]\)) given by projecting radially outward from the \(z\)-axis to the cylinder \(x^2+y^2=1\), then unrolling. The formula is \(\phi\!\left(\sqrt{1-z^2}\cos\theta, \sqrt{1-z^2}\sin\theta, z\right) = (\theta, z)\).

This is why the gnomonic projection is used in navigation charts where straight-line paths represent great-circle routes.

Stereographic Projection: The map \(\phi: S^2 \setminus \{(0,0,1)\} \to \mathbb{R}^2\) given by projecting through the north pole \((0,0,1)\) to the equatorial plane \(z=0\):

The conformality of stereographic projection means it is angle-faithful but not area-faithful. It will reappear in the Poincaré disc model: the circle inversion that defines hyperbolic reflections is a close relative of stereographic projection, and shares the same conformality property.

Chapter 3: Projective Geometry

3.1 The Projective Plane

Projective geometry arose from the study of perspective drawing. In a perspective drawing, parallel lines appear to meet at a “vanishing point” on the horizon. Projective geometry makes this rigorous by adding “points at infinity” where parallel lines meet.

We can identify \(\mathbb{P}^2\) with the set of antipodal pairs \(\{\pm\mathbf{x}\}\) in \(S^2\). A projective point is a pair of antipodal spherical points.

The key contrast with Euclidean geometry: in \(\mathbb{P}^2\), any two distinct lines always meet. There are no parallels. This is because “parallel” Euclidean lines meet at their common point at infinity.

The isometry group of \(\mathbb{P}^2\) is strictly smaller than that of \(S^2\) (\(SO(3)\) versus \(O(3)\)), because orientation-reversing maps of \(S^2\) — such as reflections — do not descend to well-defined isometries of \(\mathbb{P}^2\).

3.2 Homogeneous Coordinates and Zero Sets

To do algebra in \(\mathbb{P}^2\), we use homogeneous coordinates, which package the equivalence class \([\mathbf{x}]\) into a coordinate triple well-suited for polynomial equations.

Homogeneous polynomials are necessary because the coordinates of a projective point are only defined up to scaling: replacing \((x,y,z)\) by \((tx,ty,tz)\) changes \(F\) by \(t^n\), which vanishes if and only if \(F\) vanishes. So the zero set is consistent across representatives.

3.3 Conic Sections

The projective plane is the natural home for conic sections, where the classical distinction between ellipse, parabola, and hyperbola dissolves: all non-degenerate conics are projectively equivalent, the differences arising only from how the conic meets the line at infinity.

3.4 Classical Theorems

The projective plane is the right setting for several remarkable classical theorems about incidence — theorems concerning when points are collinear or lines are concurrent. These theorems are purely projective: they do not refer to distance or angle, only to the incidence structure of points and lines.

When the conic degenerates to a pair of lines, Pascal’s Theorem reduces to Pappus’ Theorem.

Chapter 4: Hyperbolic Geometry

4.1 Reflections in Circles

Hyperbolic geometry is built on the notion of reflection in a circle (circle inversion), which plays the role that reflection in a line plays in Euclidean geometry.

After surveying the positively curved world of the sphere and the “flat” projective plane, we now turn to the third major geometry: the hyperbolic plane, which has constant negative curvature. The construction is more subtle — we cannot simply take a surface embedded in \(\mathbb{R}^3\) as we did with \(S^2\). Instead, we will define the geometry analytically on the unit disc, and the key to understanding its isometries is circle inversion.

Geometrically, circle inversion sends points inside \(C\) to points outside \(C\) and vice versa, fixing points on \(C\) itself.

This theorem is central: it tells us that circle inversion preserves the family of “generalized circles” (circles and lines, treating lines as circles of infinite radius). This is exactly the family that will serve as hyperbolic lines, so inversions will automatically map hyperbolic lines to hyperbolic lines.

4.2 The Poincaré Disc Model

We now have all the ingredients to define the hyperbolic plane. The idea is to take the Euclidean open unit disc and equip it with a new metric that stretches distances near the boundary — so that the boundary circle is “infinitely far away” and the disc becomes a model of infinite negatively curved space.

The metric blows up at the boundary \(S^1\), meaning points near the boundary are “infinitely far away” in the hyperbolic metric. The disc looks finite to Euclidean eyes but is infinite to a hyperbolic observer.

The choice of conformal factor \(\frac{2}{1-|\mathbf{x}|^2}\) is not arbitrary: it is the unique factor (up to scaling) that makes the resulting geometry have constant curvature \(-1\). The conformality of the model — hyperbolic angles equal Euclidean angles — is what gives the Poincaré disc its aesthetic appeal in Escher’s famous “Circle Limit” drawings.

A circle \(C\) centred at \(a\) of radius \(r\) meets \(S^1\) orthogonally if and only if \(r^2 = |a|^2 - 1\) (so \(|a| > 1\)).

In both cases a hyperbolic line is an arc of a generalized circle that meets the boundary \(S^1\) at right angles — the orthogonality condition is what makes circle inversions in these circles preserve the disc.

The three-way classification of line pairs is a hallmark of hyperbolic geometry. Given a line \(L\) and a point \(p \notin L\), there are infinitely many lines through \(p\) that do not intersect \(L\) — they divide into two families of asymptotic lines bordering a continuous band of ultraparallel lines. This is the negation of Euclid’s parallel postulate.

4.3 Geodesics and Distance

The proof has the same structure as the classical Euclidean proof that straight lines are shortest: reduce to a straight radial path by discarding angular components, then integrate. The key difference is the metric weight \(\frac{2}{1-r^2}\), which makes the calculation purely one-dimensional.

The appearance of \(\cosh^{-1}\) in the distance formula, where \(\cos^{-1}\) appeared in the spherical formula, is not a coincidence: it reflects the fact that spherical and hyperbolic geometry are related by replacing trigonometric functions with their hyperbolic counterparts, corresponding to curvatures \(+1\) and \(-1\) respectively.

These formulas grow exponentially in \(r\), reflecting the negative curvature of \(\mathbb{H}^2\). For large \(r\), \(L \approx \pi e^r\) — circles grow much faster in hyperbolic space than in Euclidean space.

4.4 Angles and Triangles in Hyperbolic Geometry

Hyperbolic triangles have interior angle sum strictly less than \(\pi\), and the deficit \(\pi - (\alpha + \beta + \gamma)\) equals the area — exactly analogous to the spherical excess.

Horocycles and hypercycles are the hyperbolic analogues of circles and lines in the Euclidean sense: a horocycle is orthogonal to all hyperbolic lines through its “centre at infinity,” while a hypercycle is the locus of points at constant distance from a given hyperbolic line.

Comparing the three geometries:

The table reveals a beautiful pattern: the hyperbolic laws are obtained from the spherical laws by replacing \(\cos a\) with \(\cosh a\) and \(\sin a\) with \(\sinh a\). This is the “dictionary” between the two non-Euclidean geometries, reflecting that one has curvature \(+1\) and the other \(-1\).

This is the hyperbolic analogue of Girard’s theorem. The area equals the angle defect \(\pi - (\alpha+\beta+\gamma) > 0\), since in hyperbolic geometry \(\alpha + \beta + \gamma < \pi\). Both the spherical and hyperbolic triangle area formulas are instances of the Gauss-Bonnet theorem.

4.5 Isometries of the Hyperbolic Plane

Having classified the isometries of Euclidean space and \(S^2\), we now turn to the isometries of \(\mathbb{H}^2\). The structure is richer than in the other geometries, reflecting the more complex topology of the hyperbolic plane.

The three types of orientation-preserving isometries correspond to the three types of Möbius transformations (elliptic, parabolic, hyperbolic).

4.6 Other Models of Hyperbolic Geometry

The Poincaré disc is one of several equivalent models.

Each model of the hyperbolic plane emphasizes a different aspect of its geometry. Just as no single map projection faithfully represents the sphere, no single model of \(\mathbb{H}^2\) simultaneously preserves angles, distances, and the appearance of geodesics as straight lines. The choice of model is a matter of which properties are most useful for the task at hand.

Poincaré Upper Half-Plane Model: Let \(\mathbb{U}^2 = \{(x,y) \mid y > 0\}\). Let \(C = C_E((0,1), \sqrt{2})\) and \(L\) be the \(x\)-axis; then \(S = F_L \circ F_C\) maps \(\mathbb{H}^2\) to \(\mathbb{U}^2\). In \(\mathbb{U}^2\), the metric is \(ds = \frac{d_E s}{y}\). Geodesics are vertical lines and upper semicircles with centres on the \(x\)-axis. Angles are Euclidean.

A stereographic-like projection from \((0,0,-1)\) maps \(\mathbb{M}^2\) to \(\mathbb{H}^2\). Geodesics in \(\mathbb{M}^2\) are intersections with planes through the origin.

The Minkowski model makes explicit the parallel with spherical geometry: just as \(S^2\) is the set of unit vectors for the Euclidean inner product, \(\mathbb{M}^2\) is the set of “unit vectors” for the Minkowski inner product \(\langle (x,y,t), (x',y',t') \rangle = xx' + yy' - tt'\). The sign change in the time coordinate is precisely what switches from positive to negative curvature.

Klein Model: The unit disc \(\mathbb{K}^2 = \{(x,y) \mid x^2+y^2 < 1\}\) with a different metric defined so that the gnomic projection from \(\mathbb{M}^2\) is an isometry. In the Klein model, geodesics are Euclidean straight-line segments, but angles are not the same as Euclidean angles.

The three disc/plane models (Poincaré disc, upper half-plane, Klein) are conformally, isometrically, or projectively equivalent to each other, each highlighting different geometric properties. The Poincaré disc and upper half-plane are conformal (angle-preserving); the Klein model has straight-line geodesics. All satisfy the same abstract axioms of hyperbolic geometry, including the negation of Euclid’s parallel postulate: through any point not on a given line, there are infinitely many lines parallel to (not intersecting) the given line.