Metric and Euclidean Spaces

Metric Space

• Defined as a set with a notion of distance between its elements
• A metric space is an ordered pair (M, d) where M is a set and d is a distance on M
• d: M × M → R
• distance axioms for points x, y, z ∈ M:
  • The distance from a point to itself is zero: d(x,x) = 0
  • Distance between two points is always positive: if x ≠ y, then d(x, y) > 0
  • The distance from point x to y is always the same as the distance from y to x: d(x,y) = d(y, x)
  • The triangle inequality holds

Euclidean Space

• The fundamental space of geometry for representing physical space
• It was introduced by Euclid
• Euclidean space is a metric space where geometric objects (points, lines, planes) are defined based on axioms of Euclidean geometry
• Fundamental properties include:
  • Dimensionality: Finite dimensions, commonly 2D or 3D, possible to have n-dimensions where n is a positive integer
  • Metric: Commonly used metric is Euclidean distance. The distance between two points X1 and X2 in a plane with coordinates (x1, y1) and (x2, y2) is: d(X1,X2) = √(x1-x2)² + (У₁ - У2)2
  • Parallel Postulate: Given a line and a point not on the line, there is one parallel line that passes through the given point

Cartesian Space

• Also known as Cartesian Coordinate Space
• Type of Euclidean Space where points are specified by their coordinates relative to a set of orthogonal axes
• Introduced by René Descartes in "La Géométry" (1637)
• Coordinate of a point P(x, y) in new system (X-Y) with origin at (a, b) in the original system (X-Y):
  • x = x - a
  • ŷ= y - b
• Coordinate of a point P(x, y) in a new system (X-Ŷ) which makes an angle φ with the origin system:
  • x = x cos φ + y sin φ -ŷ= −x sin ¢ + y cos ф

Polar Coordinate System

• Two-dimensional coordinate system with a distance from a reference point (pole) and an angle from a reference direction
• Converting between Polar and Cartesian:
  • x = r cos θ
  • y = r sin θ

Linear Equations and Straight Lines

• Linear equation in one variable x has the form ax + b = 0, where a ≠ 0 and b are real numbers.
  • Its solution is x = -b/a
• Linear equation in two variables x and y has the form ax + by + c = 0, where a ≠ 0, b ≠ 0, and c are real numbers.
  • Its solution is y = -a/b x - c/b, giving a unique y value for every x.
• Slope of a line L is the ratio of change in y-direction to change in x-direction:
  • m = Δy/Δx
• Point-slope form equation of a line:
  • y - y₁ = m (x – x1), m is slope, passes through point (x1, y1)
• Slope-intercept form equation of a line:
  • y = mx + c , m is slope, c is y-intercept
  • Two lines, y = m₁ x + c₁ and y = m2 x + c2, are parallel if m₁ = m2.
  • They are perpendicular if m₁m2 = −1.
• Two-points form: y -y₁ / x-x₁ = y₂ -y₁ / x₂ -x₁, passes through two given points (X1,Y1) and (X2,Y2)
• Distance of a point P = (x1, y1) from a line ax + by + c = 0:
  • d = |ax₁ + by₁ + c| / √(a² + b²)
• Distance between two parallel lines ax + by + c₁ = 0 and ax + by + c2 = 0 :
  • d = |C1 - C2| / √(a² + b²)
• Solution for simultaneous linear equations:
  • Exists if there is a common point satisfying both equations
  • For two equations a₁x + b₁y + C₁ = 0 and a2x + b2y + C2 = 0, the slopes must not be equal: a1/b1 ≠ a2/b2, which means a1b2 ≠ a2b1.
• Solving simultaneous linear equations can be done by substitution or elimination

Linear Equations System types:

• Infinitely many solutions: n > m (number of unknowns is greater than number of equations). Known as underdetermined system.
• A single unique solution: n = m.
• No solution: n < m. Called an overdetermined system.

Quadratic Equation

• Has the form ax² + bx + c = 0
  • where a ≠ 0, b, and c are real numbers
• Roots of the quadratic are the x values that satisfy ax² + bx + c = 0
• Has one or two solutions, given by: -x= -b ± √(b² - 4ac) / 2a
• Graph of function y = f(x) = ax² + bx + c forms a parabola with an axis of symmetry parallel to the y-axis

Polynomials

• The function of form y = f(x) = anxn + An-1xn-1 + ... + a₁x + ao = Σ aᵢxⁱ
  • A degree n polynomial in variable x
  • ao, a1, …, an are real number constants and the power of the variable x are positive integers
  • a0, a1, …, an are coefficients of the polynomial
• Examples:
  • y = a₁x + ao is a linear function (1st degree polynomial)
  • y = a2x² + a₁x + ao is a quadratic function (2nd degree polynomial)
  • y = a3x³ + a2x² + a₁x + ao is a cubic function (3rd degree polynomial) -y = a4x⁴ + a3x³ + a2x² + a₁x + ao is a quartic function (4th degree polynomial)
• Root(s) of a polynomial y = f(x) are the value(s) of x that satisfy f (x) = 0
• An nth degree polynomial has at most n roots, which can include complex numbers
• Finding roots of a polynomial:
  • Factorization: low-degree polynomials, n = 2 or 3
  • Root-finding algorithm: higher-degree polynomials, n = 3 or more

Conic Sections

• Formed by the intersection of a plane with a cone
• The main section types:
  • Circle: β = 90°
  • Ellipse: a < β < 90°
  • Parabola: a = β
  • Hyperbola: 0 ≤ β < a
• A fixed vertical line l and another line m form the cone.
  • Point V is the vertex
  • Line l is axis of the cone

Circle

• Set of points in a plane equidistant from a fixed point (center)
• The distance from the center is called the radius
• Equation of a Circle (Cartesian):
  • (x − h)² + (y − k)² = r²
  • where (h, k) is the center and r is radius
• Polar Coordinate: x = r cos θ + h

Ellipse

• Set of all points where the sum of the distances from two fixed points (foci) is constant:
  • P1F1 + P1F2 = P2F1 + P2F2
• The midpoint of the foci is the center
• The line segment through the foci is the major axis
• The line segment through the center and perpendicular to the major axis is call the minor axis
• Sum of point on Ellipse (minor axis): F₁P + F2P = √b² + c² + √b² + c² = 2√b2 + c2
• Eccentricity of an ellipse is defined as: e = C / a
• The standard equation of an ellipse. x²/a² + y²/ b² = 1
• For an ellipse with a major axis parallel to the x-axis and center at (h, k): (x-h)² / a² + (y-k)²/ b² = 1
• For an ellipse with a major axis parallel to the y-axis and center at (h, k): (x-h)²/ b²+ (y-k)²/ a² = 1

Parabola

• Set of points equidistant from a fixed line (directrix) and a fixed point (focus)
• The line through the focus and perpendicular to the directrix is the axis
• Standard equation of a parabola: y² = 4ax, where a > 0
• For vertex (h, k):
  • y = p(x – h)² + k
  • x = p(y − k)² + h
• (y − k)² = 4a(x – h) opens to the right
  • (y − k)² = −4a(x – h) opens to the left
  • (x − h)² = 4a(y - k) opens upward
  • (x − h)² = −4a(y - k) opens downward

Hyperbola

• All points in a plane set where the difference of distances from two fixed points (foci) is constant
• The midpoint of the line segment joining the foci is the center
• The distance between the foci is 2c
• Distance between two vertices is 2a
• Required standard equation of hyperbola: x² /a²- y²/ b²= 1

Inequalities

• Linear inequalities involve two real numbers or algebraic expressions, related by symbols like <, >, ≤, or ≥
• Difference Between Equation and Inequality: An equation represents a line or curve in a plane whereas an inequality represents a region in a plane
• To solve simultaneous linear equations:
  • Write the equation
  • Plot the line, identify the side of the line that satisfies the inequality
  • The common region shared by the three inequalities is the solution
• Quadratic Inequalities: ax² + bx + c > 0
  • select a value for x from either case and see if that satisfies original inequality