Points, Lines, Angles

Euclid wrote the Elements 2,300 years ago. The architects of the Parthenon used those same theorems. Pixar renders Toy Story from 1 million triangles - each checked via the area formula Euclid proved in 300 BC. iPhone Face ID builds a 3D mesh of 30,000 points on a face in 50 ms. Pure geometry.

• **iPhone Face ID:** 30,000 points on the surface of a face, a 3D mesh, comparison via distances and angles. Euclid knew nothing about smartphones - but he wrote the mathematics that makes them possible.
• **Pixar and 3D rendering:** Toy Story - 1 million triangles. Every frame: ray-segment intersections, normal computation via angles, distance formula. 23 centuries after Euclid.
• **GPS and navigation:** Position on Earth is a point on a sphere. The intersection of three spheres from satellites gives coordinates. A geometric problem solved 30 times per second in everyone's pocket.

Point and Coordinate System

Geometry starts with the simplest object - a **point**. A point has no dimensions: no length, no width, no height. Only a position in space. Denoted by a capital letter: A, B, C. Deceptively simple - but a point is the foundation of the GPU that renders every Pixar frame.

To describe a point's position with numbers, use a **coordinate system**. In the plane (2D), each point has two coordinates: A(x, y). Horizontal - x (abscissa), vertical - y (ordinate). Their intersection is the origin O(0, 0). Descartes invented this in 1637, uniting geometry with algebra.

René Descartes (1637) unified geometry and algebra through coordinates. Circle x²+y²=r², line y=kx+b - geometric objects became algebraic equations. A breakthrough without which there would be no computer graphics, navigation, or machine learning.

The axes divide the plane into four **quadrants**: I (+, +), II (-, +), III (-, -), IV (+, -). In computer graphics, the screen is quadrant IV (y-axis points down). That is the first surprise when moving from math to programming.

Line, Ray, Segment

Exactly one **line** passes through two distinct points - infinite in both directions. A **ray** starts at one point and extends infinitely in one direction (a ray of light, a ray in ray casting). A **segment** is the finite portion between two points - it has a length.

Two lines in the plane: **intersecting** (one common point), **parallel** (no common points, same slope), or **coincident** (all points in common). In SQL JOIN, these are three states of two sets.

**Perpendicular lines** intersect at 90°. Notation: a ⊥ b. Condition: k₁ · k₂ = -1. In ML - vector orthogonality (dot product = 0) is the same idea, extended to n-dimensional space.

Euclid's fifth postulate: through a point not on a line, exactly one parallel passes. Lobachevsky (1830) and Riemann (1854) showed that a consistent geometry can be built by replacing it. Riemannian geometry is the foundation of general relativity. Einstein described the curvature of spacetime in exactly that framework.

Types of Angles

An **angle** is a figure formed by two rays sharing a common endpoint (vertex). The measure of an angle is how much one ray is rotated relative to the other. Units: **degrees** (°) or **radians** (rad). In programming - almost always radians: sin(), cos(), atan2() in standard libraries take radians.

**Supplementary angles** - share a side, the other sides form a straight line. Sum = 180°. **Vertical angles** - sides of one are extensions of sides of the other. Vertical angles are equal. Both properties follow from the straight angle being 180°.

**Radians** are the natural unit for angles in mathematics. 1 radian: arc length equals radius. Full rotation = 2π rad. Conversion: α_rad = α_deg × π/180. Why radians are more natural: the derivative of sin(x) = cos(x) only in radians. In degrees, a π/180 factor appears everywhere.

When two lines intersect: 4 angles, 2 pairs of vertical (equal). Perpendicular lines: all 4 angles = 90°. Parallel lines cut by a transversal: corresponding angles equal, alternate interior angles equal, co-interior angles sum to 180°. These properties make it possible to build grids of parallel streets in cities.

Measurement: Distances and Angles

Measurement in geometry: **ruler** for distances, **protractor** for angles. In analytic geometry - formulas with coordinates. In computational geometry (Pixar, games, CAD) - those same formulas, executed billions of times per second on a GPU.

For points A(x₁, y₁) and B(x₂, y₂): d(A,B) = √((x₂-x₁)² + (y₂-y₁)²). This follows from the Pythagorean theorem.

The distance from a point to a line is the length of the perpendicular. For line ax + by + c = 0 and point P(x₀, y₀): d = |ax₀ + by₀ + c| / √(a² + b²). In SVM (support vector machine) this exact distance is maximized - it is the margin from the separating hyperplane.

The distance formula is a special case of a **metric**. Other metrics: Manhattan |x₂-x₁|+|y₂-y₁| (movement on a street grid), Chebyshev max(|x₂-x₁|,|y₂-y₁|) (king's move on a chessboard). In ML, cosine distance is used to compare word vectors - angle matters more than magnitude.

Point specifies position, line - direction, angle - rotation. Distance and angle formulas: the Pythagorean theorem in coordinates. On this foundation rest triangles, circles, and all the rest of geometry. Euclid → Descartes → Pixar: three layers of one idea.

Key Ideas

• **Point** - an object without dimensions, specifying position via coordinates (x, y). Descartes connected this to algebra in 1637.
• **Line** is infinite, **ray** is a half-line with an endpoint, **segment** is a finite portion with a length. Three distinct objects, three distinct applications.
• **Angles:** acute (<90°), right (90°), obtuse (>90°), straight (180°). Perpendicularity condition: k₁ × k₂ = -1.
• **Distance and angle formulas** - the Pythagorean theorem in coordinates. Euclid → Pythagoras → iPhone Face ID: one geometry, 23 centuries.

Related Topics

Points, lines, and angles are the atoms of geometry from which everything else is built:

• Triangles — Three points, three segments, three angles - a triangle inherits all the basic concepts
• Quadrilaterals and Polygons — Parallel and perpendicular lines define rectangles and parallelograms

Вопросы для размышления

• Euclid → Pixar: one geometry, 23 centuries. What exactly in Euclid's 5 postulates turned out to be so fundamental that it holds to this day?
• iPhone Face ID uses 30,000 points to recognize a face. Which geometric operations - distances, angles, normals - are used to compare two point meshes?
• Manhattan distance |x₂-x₁|+|y₂-y₁| vs Euclidean √((x₂-x₁)²+(y₂-y₁)²). When is the first one better? Which would GPS choose for city streets?

Связанные уроки

• trig-01