Analytic Geometry

A vector is a directed segment characterized by its length (magnitude) and direction. Two vectors are equal if they are parallel and of equal length—regardless of their initial point (free vectors).

Cartesian coordinates: a vector $\mathbf{a} = (a_1, a_2, a_3)$ is defined by its projections onto the axes. Magnitude $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.

Addition: $\mathbf{a} + \mathbf{b} = (a_1 + b_1, a_2 + b_2, a_3 + b_3)$. Geometrically: parallelogram rule or triangle rule.

Multiplication by a scalar: $\lambda \mathbf{a} = (\lambda a_1, \lambda a_2, \lambda a_3)$. For $\lambda > 0$—direction is preserved; for $\lambda < 0$—direction is reversed.

General equation: Ax + By + Cz + D = 0. The vector (A, B, C) is the normal vector n of the plane.

Normal equation: n·(r − r₀) = 0, where r₀ is a point of the plane, n is the normal.

Through three points: the 3×3 determinant composed of the differences with the first point equals zero.

Distance from point M(x₀, y₀, z₀) to the plane: d = |Ax₀ + By₀ + Cz₀ + D| / √(A²+B²+C²).

$a$ — major semi-axis, $b$ — minor. Foci: $F_1(-c, 0)$, $F_2(c, 0)$, $c^2 = a^2 - b^2$.

Definition: the locus of points, the sum of distances from which to two foci is constant $= 2a$.

Hyperboloids: one-sheeted ($x^2/a^2 + y^2/b^2 - z^2/c^2 = 1$), two-sheeted ($x^2/a^2 + y^2/b^2 - z^2/c^2 = -1$).

Paraboloids: elliptic ($z = x^2/a^2 + y^2/b^2$), hyperbolic (saddle: $z = x^2/a^2 - y^2/b^2$).

Affine transformation: $r' = Ar + b$ ($A$ is a non-singular matrix, $b$ is a vector).

Preserves: straightness, parallelism, ratios of lengths on parallel lines, area (with a coefficient $|\det A|$).

The group of affine transformations of the plane: $3\times3$ matrices of the form $\begin{bmatrix}A & b \\ 0 & 1\end{bmatrix}$ (homogeneous coordinates).

When rotating the axes by an angle $\varphi$: $x = x'\cos\varphi - y'\sin\varphi$, $y = x'\sin\varphi + y'\cos\varphi$.

A point on the plane: (r, φ), r ≥ 0, φ ∈ [0, 2π). Relation to Cartesian coordinates: x = r cosφ, y = r sinφ.

Curves: r = a (circle), φ = α (ray), r = aφ (Archimedean spiral), r = a(1+cosφ) (cardioid).

Conic sections in polar coordinates: r = p/(1−e cosφ), where e is the eccentricity. A single formula for the ellipse (e<1), parabola (e=1), hyperbola (e>1). Planetary orbits take exactly this form — Kepler's First Law.

Equation of a sphere: ρ = R. Application in quantum mechanics: spherical harmonics — wave functions of the hydrogen atom in spherical coordinates.

Classification: by rotating and shifting the coordinates, we bring it to canonical form (using eigenvectors of the quadratic part).

Invariants: I₁ = tr A (sum of eigenvalues), I₂ = sum of 2×2 minors, I₃ = det A — do not depend on the choice of coordinate system.

A one-sheeted hyperboloid is a surface of revolution of a hyperbola. Any generator is a straight line (ruled surface). Used in architecture (cooling towers, Shukhov tower).

Stereographic projection maps the sphere onto the extended complex plane ℂ ∪ {∞}. Conformal automorphisms of the sphere are fractional linear transformations.

Apollonius of Perga (3rd century BC) systematically studied "conic sections"—intersections of a cone with a plane. Depending on the angle of inclination of the plane, we obtain an ellipse, a parabola, or a hyperbola.

Two millennia later, Kepler discovered: planets move along ellipses, comets—along parabolas or hyperbolas. Thus, ancient geometry turned out to be a law of nature.

Ellipse: a ray originating from one focus, after reflection from the ellipse, passes through the other focus. Whispering rooms (elliptical ceilings), lithotripters for crushing kidney stones—applications of this property.

Parabola: a ray parallel to the axis, after reflection, passes through the focus. And conversely: a source located at the focus gives a parallel beam. Parabolic mirrors of telescopes, spotlights, antennas.

Formulas

Step 1: Find the eigenvalues of the matrix of the quadratic part $A_q = \begin{bmatrix} A & B/2 \\ B/2 & C \end{bmatrix}$. The characteristic polynomial: $\lambda^2 - (A+C)\lambda + (AC - B^2/4) = 0$.

Step 2: By rotation (along the eigenvectors), eliminate the $xy$ term. Angle: $\tan 2\varphi = \dfrac{B}{A-C}$.

Step 3: By shifting the coordinates (completing the square), eliminate the linear terms.

Step 4: By the signs of the coefficients at the squares, determine the type.

Cubic curves: $Ax^3 + Bx^2y + Cxy^2 + Dy^3 + \ldots = 0$. Examples: Descartes' cubic, Bernoulli's lemniscate ($r^2 = a^2 \cos 2\theta$), strophoid.

Lemniscate: $(x^2 + y^2)^2 = a^2(x^2 - y^2)$. Area $S = a^2$. Connection with elliptic integrals: the length of the lemniscate is expressed through an elliptic integral of the first kind — historically, this motivated Gauss's theory.

Cycloid: $x = R(t - \sin t)$, $y = R(1 - \cos t)$ — the trajectory of a point on a rolling circle.

The cycloid is the solution to the brachistochrone problem (the path of shortest descent time under gravity): Johann Bernoulli in 1696. The problem led to the development of the calculus of variations.

A ruled surface is one through each point of which passes a straight line lying entirely on the surface. These lines are called generators.

Cylinders and cones are obviously ruled. More surprisingly, the one-sheeted hyperboloid and the hyperbolic paraboloid are also ruled.

Two families of generators: through every point pass two lines from different families. Any two generators from one family are skew.

Constructive significance: hyperbolic towers are built from straight bars (material savings + rigidity). The Shukhov Tower in Moscow (1922) was the first example.

We add a “plane at infinity” to $\mathbb{R}^3$: $\mathbb{R}P^3$. Points are nonzero vectors $(x:y:z:w)$ in homogeneous coordinates (up to scale).

The intersection of two distinct planes is a line. In projective geometry: any two planes intersect (parallel planes intersect at infinity).

Perspective projection is a projective transformation. A pinhole camera projects a 3D scene onto a sensor (plane): $(X, Y, Z) \mapsto (fX/Z, fY/Z)$, where $f$ is the focal length.

Camera calibration = finding the parameters of the projective transformation. This is the basis of 3D reconstruction and augmented reality.

Formulas

Curvature: κ = |dτ/ds| — a measure of how the direction of the tangent changes.

{τ, ν, β} — Frenet frame — a right-handed orthonormal triple at each point of the curve.

Torsion: χ = −dβ/ds · ν — a measure of the curve "departing" from the osculating plane (the plane of τ, ν).

Theorem: a curve is determined (up to a movement) by the pair of functions κ(s) > 0 and χ(s) (the natural equations).

In 1872, Felix Klein proposed a revolutionary perspective on geometry: each geometry is determined by its group of transformations and studies the invariants of this group.

Motion = isometry: transformation preserving distances. In $\mathbb{R}^2$: rotations, reflections, parallel translations.

Proper motions (preserving orientation): rotations and translations. Form a subgroup. Improper (changing orientation): reflections and glide reflections.

Invariant under projective transformations. For special choices: harmonic division $(A,B;C,D) = -1$.

The symplectic form on $\mathbb{R}^{2n}$: $\omega = \sum_i dp_i \wedge dq_i$ is a nondegenerate skew-symmetric bilinear form.

Symplectic geometry studies spaces with such a form. The key property: $\omega$ is nondegenerate (determinant of the form’s matrix $\neq 0$) and closed ($d\omega = 0$).

The phase space $(p, q)$ of a mechanical system is a symplectic space. The Hamilton equations: $\dot{q}_i = \partial H / \partial p_i$, $\dot{p}_i = -\partial H / \partial q_i$ preserve the symplectic form (Liouville’s theorem: the volume in phase space is preserved).

The energy $H$ is conserved along a trajectory (if it does not explicitly depend on time).

Newtonian mechanics assumes Euclidean space: an absolute three-dimensional space whose metric does not depend on the observer. Time is absolute and independent.

Distances and angles are physically meaningful invariants. Newton’s laws are invariant under Galilean transformations: r' = r − vt, t' = t.

Minkowski space-time: $\mathbb{R}^4$ with the metric $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$.

Invariants: the interval $ds^2 = \text{const}$ under Lorentz transformations. Not a sum of squares, but a difference—a hyperbolic metric.