Geometry Basic

Q: What does Basic Geometry cover?

It covers university year 1-2 content: the axiom system of Euclidean geometry, the group structure of congruence and similarity transformations (transformation geometry), affine spaces and affine transformations, the basics of projective spaces and projective transformations, and the description of curves and surfaces by parametrization.

Q: What prior knowledge is needed to study Basic Geometry?

The content of Geometry Introduction (trigonometric ratios, trigonometric functions, the coordinate plane, vectors), the basics of linear algebra (matrices, linear transformations), and the basic concepts of sets and maps.

Q: What is transformation geometry?

Transformation geometry is a branch of geometry that reconsiders figures through transformations such as translation, rotation, and reflection. Using the fact that the composition of transformations forms a group, it focuses on the invariants of figures (properties unchanged by transformations) to understand their essence. It is the foundation of Klein's ideas in the Erlangen program.

Euclidean Geometry and the Viewpoint of Transformations (University Year 1-2 Level)

Overview of the Basic Level

At the basic level, classical Euclidean geometry is studied axiomatically, and figures are reconsidered from the viewpoint of transformation geometry. It also covers an introduction to projective geometry and the rudiments of curves and surfaces.

Learning goals

Understand the axiom system of Euclidean geometry
Understand the group structure of congruence and similarity transformations
Learn the basics of projective spaces and projective transformations
Master the description of curves and surfaces by parametrization

Euclidean Geometry
Axioms, congruence, similarity, properties of circles
Transformation Geometry
Translation, rotation, reflection, the group of congruence transformations
Affine Geometry
Affine spaces, affine transformations, barycentric coordinates
Introduction to Projective Geometry
Projective spaces, homogeneous coordinates, projective transformations
Curves and Surfaces
Parametrization, quadratic curves and surfaces, implicit representation

Topic Articles

Conic Section
A unified understanding of quadratic curves: discriminant, eccentricity, polar form, applications
Parabola
The quadratic curve defined by a focus and a directrix: standard form, tangents, reflective property, applications
Ellipse
Foci, eccentricity, directrix, parametrization, area and circumference, the law of reflection, applications
Hyperbola
Foci, asymptotes, eccentricity, parametrization, tangents, reflective property, applications
Ellipsoid
The three-dimensional quadric surface x²/a²+y²/b²+z²/c²=1, volume 4/3πabc, surface-area approximations, relation to oblate and prolate spheroids
Hyperboloid
Standard forms of one- and two-sheeted hyperboloids, the asymptotic cone, and the ruled-surface property of containing straight lines
Paraboloid
Standard forms of elliptic and hyperbolic paraboloids, reflective property, the geometry of saddle points
Oblate Spheroid
The spheroid of revolution with a=b>c: volume and surface-area formulas, applications such as the shape of the Earth
Prolate Spheroid
The spheroid of revolution with a=b<c: volume and surface-area formulas, applications
Cylindrical Coordinates
The definition of cylindrical coordinates, conversion formulas with Cartesian coordinates, the volume element, and applications

Reading

Another Ending for Parallel Lines — The World That Opened by Doubting a Postulate [Reading]
The two-thousand-year puzzle of the parallel postulate. A relaxed look at the birth of non-Euclidean geometry, discovered by Bolyai and Lobachevsky.
What Ruler and Compass Can and Cannot Build — The Story of the Three Classical Construction Problems [Reading]
Why can't a general angle be trisected? A relaxed telling of how the three classical construction problems, attempted for two thousand years, were finally proved impossible.
How Many Geometries Are There? — The Erlangen View That Classifies by What Stays Unchanged [Reading]
Why are there several geometries? Klein saw that which transformations leave things unchanged is what decides a geometry. A relaxed look at organizing geometry through transformation groups and invariants.

Prerequisites

The content of Geometry Introduction
Basics of linear algebra (matrices, linear transformations)
Basic concepts of sets and maps

The Viewpoint of Transformations

Transformations capture the essential properties (invariants) of a figure.

References

Frequently Asked Questions

Q1: What does Basic Geometry cover?

A: It covers university year 1-2 content: the axiom system of Euclidean geometry, the group structure of congruence and similarity transformations (transformation geometry), affine spaces and affine transformations, the basics of projective spaces and projective transformations, and the description of curves and surfaces by parametrization.

Q2: What prior knowledge is needed to study Basic Geometry?

A: The content of Geometry Introduction (trigonometric ratios, trigonometric functions, the coordinate plane, vectors), the basics of linear algebra (matrices, linear transformations), and the basic concepts of sets and maps.

Q3: What is transformation geometry?

A: Transformation geometry is a branch of geometry that reconsiders figures through transformations such as translation, rotation, and reflection. Using the fact that the composition of transformations forms a group, it focuses on the invariants of figures (properties unchanged by transformations) to understand their essence. It is the foundation of Klein's ideas in the Erlangen program.