Intro To Geometry // Flashcards
5 axioms of Euclid
- Through any two points there passes a unique line
- It is possible to extend any line segment continuously in a straight line to a larger line segment
- It is possible to draw a unique circle of any given radius around any given point
- All right angles are equal to each other
- (Parallel postulate) if a straight line crossing two straight lines makes interior angles on one of the sides which are less than two right angles in total, then the two straight lines, if produced indefinitely, meet on that side
Angle from a to b
Given two intersecting lines a and b, we denote by ∠ab the value of either of the two equal angles where we go from a to b anti-clockwise
Perpendicular
Two lines a and b are said to be perpendicular (a ⊥ b) if they intersect at right angles, that is ∠ab = ∠ba = π/2
Map (geometry)
A map from the plane to the plane is a rule that sends each point in the plane to some other point in the plane. Given a map f and a point P in the plane, we denote f(P) the point where f sends P
Perpendicular bisect
Given any line segment AB its perpendicular bisect is the unique line which passes through the midpoint of AB and is perpendicular to AB
Reflection
The reflection r_l in l is the map which sends any point P to the unique point r_l(P) such that l is the perpendicular bisect of the line segment Pr_l(P)
Dilation D_O,α
Given a point in the plane define the dilation D_O,α with centre O and scale factor α to be the map which leaves the point O fixed and sends any point P ≠ O to the unique point D_O,α(P) which lies on the continuation of the line segment OP in the direction of P at the distance α|OP| from O
Translation T_v
Given a vector v = AB (oriented line segment), define the translation T_v by v to be the map which translates every point in the plane by AB
Isometry
A map f: Plane → Plane is an isometry if for any two points A and B in the plane we have d(A,B) = d(f(A),f(B))
Rotation R_O,θ
Given a point O in the plane the rotation R_O,θ with centre O and angle 0 ≤ θ < 2π is the map such that for every point P ≠ O, R_O,θ(P) is the unique point P’ on the circle with centre O and radius |OP| such that ∠POP’ = θ
Orientation preserving
An isometry f: Plane → Plane is orientation preserving if for any three points A, B and C in the plane we have ∠ABC = ∠f(A)f(B)f(C)
Orientation reversing
An isometry f: Plane → Plane is orientation reversing if for any three points A, B and C in the plane we have ∠ABC = ∠f(C)f(B)f(A)
Congruent
Two geometrical figures are congruent if there exists an isometry taking one to the other, denoted by ≅. If a vertex order is specified it must be preserved, e.g. ΔABC ≅ ΔA’B’C’ requires that there exists an isometry taking A to A’, B to B’ and C to C’
Clockwise orientated triangle
A triangle ΔABC is clockwise orientated if moving from A to B to C takes you clockwise around the points in the interior of the triangle
3 criteria for congruence of ΔABC and ΔA’B’C’
- (SAS) |BA|=|B’A’|, |BC|=|B’C’| and β=β’
- (ASA) |BC|=|B’C’|, β=β’ and γ=γ’
- (SSS) |AB|=|A’B’|, |BC|=|B’C’| and |CA|=|C’A’|
Tangent
A line l is tangent to a circle C at point P if it meets C only at P
Distance from P to ℓ
The distance from a point P to a line ℓ is 0 if P ∈ ℓ, and is |OP| for the unique O ∈ ℓ with PO ⊥ ℓ, otherwise
Great circle/geodesic
A great circle on a sphere is the intersection of this sphere with a plane passing through its centre
Antipodal
Two points on a sphere are called antipodal if they lie on opposite ends of a diameter of the sphere
Ambient distance (3 space)
Distance from A to B is the length of the straight line joining them
Intrinsic distance
The distance from A to B is the length of the smaller of the two arcs of any great circle joining A and B
Tangent plane
Let S be the surface of a sphere in a 3D space, let O be its centre and P ∈ S a point. Then the plane through P, perpendicular to OP, is the unique plane tangent to S at P
Spherical angle
Let AB and AC be two arcs of two great circles ℓ_B and ℓ_C on a sphere S. Let π_A be the plane tangent to S at A, and let t_B and t_C be the tangent lines carved out on π_A by the planes of ℓ_B and ℓ_C respectively. The spherical angle ∠BAC between ℓ_B and ℓ_C is defined to be the planar angle ∠t_B,t_C between t_B and t_C in plane π_A
Parallel Postulate
If a straight line crossing two straight lines makes the interior angles on one of the sides which are less than two right angles in total, then the two straight lines, if produced indefinitely, meet on that side
Playfair’s axiom
Given a line ℓ and a point P not on ℓ there is a unique line through P parallel to P
3 axioms defining area
- The area of a rectangle is the product of the lengths of its two adjacent sides
- If two figures are disjoint or meet only along their edges, then the area of their union is the sum of their areas
- Congruent figures have equal areas
Pythagoras’ theorem
Let ΔABC be a triangle in the plane such that B is a right angle, then |AB|^2 + |BC|^2 = |AC|^2
Sum of angles in a triangle ΔABC in a sphere
∠A + ∠B + ∠C = π + Area(ΔABC)/R², where R = radius of sphere
Stereographic projection
A function F: S{p_n} → ∏, where ∀ A ∈ S, p_nA ⊂ E^3 is the unique straight line passing through A and p_n, and F(A) ∈ ∏ is the unique intersection point of p_nA and ∏
Inversion wrt a circle C₀
The map I_C₀: ∏ \ {O} → ∏ \ {O}, defined by I_C₀(P) = F(refl_∏(F⁻¹(P))) for every P ∈ ∏ \ {O}, and refl_∏ is the reflection with respect to ∏
