The Central Idea
Geometry depends partly on the structure of the space in which geometric objects live.
In a flat Euclidean plane:
- straight lines behave in familiar ways;
- parallel lines remain a constant distance apart;
- triangle angle sums equal \(180^\circ\);
- the Pythagorean theorem has its familiar form.
In curved spaces, some of these relationships change.
Non-Euclidean geometry studies spaces in which Euclidean rules are modified because the underlying geometry is different.
What Is a Geometric Space?
A geometric space is a mathematical setting in which positions, distances, directions, curves, and other spatial relationships can be defined.
A geometric space may be:
- one-dimensional;
- two-dimensional;
- three-dimensional;
- higher-dimensional;
- flat;
- curved;
- finite or infinite in extent.
Euclidean Geometry
Euclidean geometry is the geometry associated with the classical work of Euclid and the familiar geometry of flat space.
It describes idealized points, lines, planes, angles, polygons, circles, solids, and their relationships under a particular system of axioms and postulates.
The Euclidean Plane
The ordinary Cartesian coordinate plane
\[ \mathbb{R}^2 \]is a standard model of Euclidean two-dimensional space.
Its points are ordered pairs
\[ (x,y), \]where
\[ x,y\in\mathbb{R}. \]Euclidean Three-Dimensional Space
Ordinary three-dimensional Cartesian space is represented by
\[ \mathbb{R}^3. \]Its points have coordinates
\[ (x,y,z). \]This provides the standard mathematical model for much of elementary solid geometry.
Euclidean Distance
In the Euclidean plane, the distance between
\[ A=(x_1,y_1) \]and
\[ B=(x_2,y_2) \]is
\[ \boxed{ d(A,B)= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }. \]This formula comes from the Pythagorean theorem.
Euclidean Lines
In the Euclidean plane, a line is a straight one-dimensional object extending indefinitely in both directions.
A nonvertical line can be represented by
\[ y=mx+b. \]Its slope \(m\) describes direction.
Euclid's Parallel Postulate
One of the most important features of Euclidean geometry concerns parallel lines.
In a common modern form:
Through a point not on a given line, exactly one line can be drawn parallel to the given line.
This statement distinguishes Euclidean geometry from major non-Euclidean geometries.
Parallel Lines in Euclidean Space
Distinct parallel lines in a Euclidean plane:
- are coplanar;
- do not intersect;
- share the same direction;
- remain a constant perpendicular distance apart.
These familiar properties depend on the flat structure of Euclidean space.
Euclidean Triangle Angle Sum
In Euclidean geometry, the interior angles of every triangle satisfy
\[ \boxed{ A+B+C=180^\circ }. \]This familiar result is not universal across all geometries.
Euclidean Geometry Is Flat
Euclidean geometry is often described as flat geometry.
A plane has zero intrinsic curvature.
Locally and globally, its straight-line geometry follows the standard Euclidean rules.
What Is Non-Euclidean Geometry?
Non-Euclidean geometry studies geometric systems in which one or more Euclidean assumptions, especially the parallel postulate, do not hold in the usual form.
Two major examples are:
- spherical geometry;
- hyperbolic geometry.
These geometries are logically consistent and mathematically useful.
Curvature Changes Geometry
One useful way to compare geometric spaces is through curvature.
- Euclidean space has zero curvature.
- Spherical geometry has positive curvature.
- Hyperbolic geometry has negative curvature.
Curvature affects distance, shortest paths, triangle angle sums, and parallel behavior.
Positive Curvature
A sphere provides the standard intuitive model of positive curvature.
The surface bends consistently in the same general sense.
Geodesics that begin in different directions can eventually meet again.
Negative Curvature
Hyperbolic space has negative curvature.
A saddle-shaped surface provides a useful local visual analogy, although no ordinary finite saddle surface reproduces the complete hyperbolic plane.
Geodesics tend to diverge more strongly than they do in Euclidean space.
Geodesics
A geodesic is the natural generalization of a straight line to a curved space.
Informally, a geodesic is a locally shortest or locally straight path within the geometry of the space.
In a Euclidean plane, geodesics are ordinary straight lines.
Great Circles
On a sphere, geodesics are portions of great circles.
A great circle is formed by intersecting a sphere with a plane passing through the sphere's center.
Examples include the equator and any complete meridian circle on an idealized spherical Earth.
Great Circles Versus Smaller Circles
Not every circle drawn on a sphere is a geodesic.
A circle of latitude other than the equator is generally not a great circle.
Only circles whose defining plane passes through the center of the sphere are great circles.
Shortest Paths on a Sphere
Between two sufficiently separated points on a sphere, the shortest route along the surface follows an arc of a great circle.
This is why long-distance navigation often uses great-circle routes rather than paths that appear straight on a flat map.
Spherical Geometry
Spherical geometry studies geometric relationships on the surface of a sphere.
Points are points on the sphere.
The role played by straight lines is taken by great circles.
Parallel Lines Behave Differently on a Sphere
Distinct great circles always intersect.
Therefore, in spherical geometry there are no parallel great-circle geodesics in the Euclidean sense.
Through a point not on a given great circle, there is no great circle that remains disjoint from the original one.
Great Circles Intersect Twice
Any two distinct great circles intersect at two antipodal points.
For example, two different meridian circles meet at the north and south poles.
This is very different from the Euclidean rule that two distinct lines meet at most once.
Spherical Triangles
A spherical triangle is formed by three great-circle arcs.
Its sides are geodesic arcs rather than ordinary Euclidean line segments.
Spherical triangles obey different angle-sum rules.
Spherical Triangle Angle Sum
In spherical geometry, the angle sum of a triangle is greater than
\[ 180^\circ. \]Thus,
\[ \boxed{ A+B+C>180^\circ }. \]The excess above \(180^\circ\) is related to the triangle's area on the sphere.
Example: A Spherical Triangle
Consider a spherical triangle formed by:
- the north pole;
- one point on the equator;
- another point on the equator \(90^\circ\) of longitude away.
Each of the three interior angles can measure
\[ 90^\circ. \]The total is
\[ 90^\circ+90^\circ+90^\circ = \boxed{270^\circ}. \]Spherical Distance
On a sphere, distance is measured along the surface rather than through the interior.
If two points subtend a central angle \(\theta\) measured in radians on a sphere of radius \(R\), the great-circle arc length is
\[ \boxed{s=R\theta}. \]Chord Distance and Surface Distance Are Different
Two points on a sphere can be connected by a straight chord through three-dimensional space or by an arc along the spherical surface.
These are different distances.
Spherical geometry uses the intrinsic surface distance.
Hyperbolic Geometry
Hyperbolic geometry is a non-Euclidean geometry with negative curvature.
It modifies the Euclidean parallel postulate in the opposite direction from spherical geometry.
Parallel Behavior in Hyperbolic Geometry
In hyperbolic geometry, through a point not on a given geodesic, more than one geodesic can pass without intersecting the given geodesic.
In fact, infinitely many such geodesics exist.
This differs sharply from the unique Euclidean parallel.
Hyperbolic Triangle Angle Sum
In hyperbolic geometry, the interior angle sum of a triangle is less than
\[ 180^\circ. \]Thus,
\[ \boxed{ A+B+C<180^\circ }. \]The deficit below \(180^\circ\) is related to the triangle's area.
Triangle Angle Sum as a Curvature Signal
Triangle angle sums provide a useful comparison among the three geometries.
- Euclidean: \(A+B+C=180^\circ\).
- Spherical: \(A+B+C>180^\circ\).
- Hyperbolic: \(A+B+C<180^\circ\).
These differences reflect the curvature of the underlying space.
Models of Hyperbolic Geometry
Hyperbolic geometry is often represented using models drawn inside ordinary Euclidean space.
Common models include:
- the Poincaré disk model;
- the Poincaré half-plane model;
- the Klein disk model.
These drawings represent hyperbolic relationships using Euclidean coordinates.
The Poincaré Disk Model
In the Poincaré disk model, the entire hyperbolic plane is represented inside a Euclidean disk.
Hyperbolic geodesics appear as:
- diameters of the disk;
- circular arcs meeting the boundary circle at right angles.
The outer circle represents infinitely distant points and is not part of the hyperbolic plane itself.
A Model Is Not the Geometry Itself
A mathematical model translates one geometry into another representational framework.
A curved-looking arc in a Euclidean drawing may represent a hyperbolic geodesic.
Therefore, visual straightness in the surrounding page is not the same as intrinsic straightness in the modeled geometry.
Intrinsic and Extrinsic Geometry
Intrinsic geometry describes measurements and relationships that can be determined within the space itself.
Extrinsic geometry describes how that space is embedded or situated inside a larger space.
A sphere's surface has intrinsic curvature even though we often visualize it as embedded in three-dimensional Euclidean space.
A Two-Dimensional Being on a Surface
Imagine a being restricted to moving on a surface.
It could measure distances, angles, and triangle sums without leaving the surface.
Those measurements could reveal whether its space is flat or curved.
This illustrates the meaning of intrinsic geometry.
Local and Global Geometry
A curved space can look approximately Euclidean when examined over a very small region.
The surface of Earth appears nearly flat over short distances because Earth's radius is large compared with everyday scales.
Over larger distances, curvature becomes significant.
Maps Distort Curved Surfaces
A flat map attempts to represent part or all of a curved surface in a Euclidean plane.
No flat map of a sphere can preserve every geometric property globally.
Different map projections preserve different combinations of:
- angles;
- areas;
- distances;
- directions;
- shapes.
Euclidean Geometry Remains Extremely Useful
Non-Euclidean geometry does not make Euclidean geometry obsolete.
Euclidean geometry is an excellent model whenever curvature is negligible at the scale of interest.
Engineering drawings, buildings, local land measurements, and many physical models can often be treated accurately with Euclidean methods.
Choosing the Right Geometry
The appropriate geometry depends on the problem.
For example:
- a small room can usually be modeled Euclideanly;
- global airline routes benefit from spherical geometry;
- certain theoretical spaces are modeled hyperbolically;
- general relativity uses curved spacetime geometry.
Geometry is therefore a modeling choice as well as an abstract theory.
Euclidean and Non-Euclidean Geometry Share Core Ideas
These geometries differ in important rules, but they share broad structural ideas.
Each can study:
- points;
- paths or geodesics;
- distance;
- angles;
- triangles;
- symmetry;
- transformations.
The definitions and resulting relationships adapt to the geometry of the space.
The Meaning of Straight Depends on the Space
In Euclidean geometry, straight lines are familiar linear paths.
On a sphere, great-circle arcs play the role of straightest possible paths.
In hyperbolic geometry, geodesics follow the intrinsic structure of hyperbolic space.
“Straight” is therefore a geometric property relative to a space, not merely a visual property of a drawing.
The Meaning of Parallel Depends on the Space
The number of parallels through a point not on a given geodesic provides a classic comparison.
- Euclidean geometry: exactly one.
- Spherical geometry: none.
- Hyperbolic geometry: more than one.
This is one of the clearest demonstrations that different consistent geometries can exist.
The Meaning of Distance Depends on the Space
Euclidean distance uses straight-line separation in flat space.
Spherical distance follows great-circle arcs.
Hyperbolic distance uses the metric of hyperbolic space.
Changing the metric changes what counts as the shortest route.
Metrics
A metric is a rule for measuring distance.
In Euclidean geometry, the familiar distance formula defines the standard Euclidean metric.
Other spaces can use different metrics while still satisfying the general requirements expected of distance.
Higher-Dimensional Euclidean Spaces
Euclidean geometry is not limited to two or three dimensions.
The space
\[ \mathbb{R}^n \]generalizes Cartesian coordinates to \(n\) dimensions.
A point has coordinates
\[ (x_1,x_2,\ldots,x_n). \]Distance in Higher-Dimensional Euclidean Space
For points
\[ A=(a_1,\ldots,a_n) \]and
\[ B=(b_1,\ldots,b_n), \]the Euclidean distance is
\[ \boxed{ d(A,B) = \sqrt{ \sum_{i=1}^{n}(b_i-a_i)^2 } }. \]The familiar plane and three-dimensional formulas are special cases.
Curved Spaces Can Also Have Higher Dimensions
Curvature is not restricted to two-dimensional surfaces.
Mathematicians study curved spaces of many dimensions.
The language of manifolds and differential geometry provides a systematic framework for these spaces.
This lesson only introduces the central spatial idea.
Geometry and Physics
Modern physics uses non-Euclidean geometry to model physical space and spacetime.
In general relativity, gravity is described through curvature of spacetime rather than simply as a force acting in ordinary flat space.
This is one of the most important applications of curved geometry.
Geometry and Navigation
Navigation across Earth provides an accessible application of spherical ideas.
Over short distances, a flat map may work well.
Over global distances, great-circle geometry better captures shortest routes along Earth's surface.
Geometry and Computer Graphics
Computer graphics uses many kinds of geometric spaces and coordinate systems.
Curved surfaces may be represented locally with Euclidean coordinates and then assembled into larger models.
Transformations convert between local, world, camera, and screen coordinate spaces.
Common Error: Assuming Euclidean Geometry Is the Only Geometry
Euclidean geometry is familiar and extremely useful, but it is one geometric system among many.
Non-Euclidean geometries are not errors or distorted versions of Euclidean geometry. They are internally consistent systems with different structural assumptions.
Common Error: Treating the Surface of a Sphere as a Euclidean Plane
Small regions of a sphere may appear nearly flat.
Globally, however, spherical geometry differs from Euclidean geometry.
Great circles intersect, and triangle angle sums can exceed \(180^\circ\).
Common Error: Calling Every Circle on a Sphere a Geodesic
Only great circles are spherical geodesics.
Smaller circles, such as most circles of latitude, are not locally straight paths in spherical geometry.
Common Error: Assuming Triangle Angles Always Sum to \(180^\circ\)
The \(180^\circ\) rule is Euclidean.
Spherical triangles have larger angle sums, while hyperbolic triangles have smaller angle sums.
Always identify the geometry before applying the rule.
Common Error: Assuming Parallel Means the Same Thing Everywhere
Parallel behavior depends on the geometry.
Euclidean, spherical, and hyperbolic geometries have different answers to the parallel question.
Common Error: Confusing the Model with the Space
A Poincaré disk drawing lies in a Euclidean page, but it represents hyperbolic geometry.
Curved Euclidean arcs in the drawing can represent hyperbolically straight geodesics.
The geometry must be interpreted according to the model's rules.
Common Error: Assuming Curvature Must Be Visually Obvious
Curvature can be detected intrinsically through measurements such as triangle angle sums and distance relationships.
A space may appear almost flat locally while differing significantly from Euclidean geometry at larger scales.
A Reliable Comparison Procedure
When working in an unfamiliar geometry, ask:
- What space is being used?
- What counts as a straight path or geodesic?
- How is distance measured?
- What is the curvature?
- How do parallel geodesics behave?
- What is the triangle angle-sum rule?
- Is the diagram intrinsic to the geometry or only a model drawn in another space?
- Which Euclidean results remain valid and which must be modified?
A Compact Comparison
| Geometry | Curvature | Parallels through external point | Triangle angle sum |
|---|---|---|---|
| Euclidean | Zero | Exactly one | \(180^\circ\) |
| Spherical | Positive | None | Greater than \(180^\circ\) |
| Hyperbolic | Negative | More than one | Less than \(180^\circ\) |
Practice 1: Euclidean Triangle
Two angles of a Euclidean triangle are
\[ 50^\circ \]and
\[ 60^\circ. \]Find the third angle.
Practice 2: Identify the Geometry
A triangle has angle sum
\[ 210^\circ. \]Which of the three geometries introduced in this lesson is consistent with that result?
Practice 3: Hyperbolic Triangle
A triangle has angle sum
\[ 160^\circ. \]Which geometry is consistent with that result?
Practice 4: Great Circle
Is the equator of a sphere a great circle?
Practice 5: Circle of Latitude
Is every circle of latitude on a sphere a geodesic?
Practice 6: Spherical Distance
Two points on a sphere of radius \(10\) subtend a central angle of
\[ \frac{\pi}{3}. \]Find the great-circle arc length between them.
Practice 7: Parallel Postulate
In Euclidean geometry, how many lines through a point not on a given line are parallel to that line?
Practice 8: Spherical Parallels
In spherical geometry using great circles as geodesics, how many geodesics through a point remain disjoint from a given great circle?
Practice 9: Hyperbolic Parallels
In hyperbolic geometry, is there exactly one geodesic through an external point that avoids a given geodesic?
Practice 10: Identify the Error
A student says, “Because the shortest path between two points is always a straight line, a great-circle route on Earth cannot be a shortest path.”
Explain the error.
Practice Answers
Practice 1
\[ 180^\circ-50^\circ-60^\circ = \boxed{70^\circ}. \]Practice 2
Since
\[ 210^\circ>180^\circ, \]the result is consistent with spherical geometry.
Practice 3
Since
\[ 160^\circ<180^\circ, \]the result is consistent with hyperbolic geometry.
Practice 4
Yes. The equator lies in a plane passing through the center of the sphere, so it is a great circle.
Practice 5
No. Only great circles are spherical geodesics. Most circles of latitude are smaller circles whose defining planes do not pass through the center of the sphere.
Practice 6
Use
\[ s=R\theta. \]Therefore,
\[ s= 10\left(\frac{\pi}{3}\right) = \boxed{\frac{10\pi}{3}}. \]Practice 7
Exactly
\[ \boxed{1}. \]Practice 8
None. Distinct great circles always intersect.
Practice 9
No. Hyperbolic geometry allows more than one such geodesic; in fact, infinitely many geodesics through the external point can avoid the given geodesic.
Practice 10
“Straight” must be interpreted within the geometry of the space. On a sphere, great-circle arcs are geodesics and play the role of straightest surface paths. A route can look curved on a flat map while being a shortest path on the spherical surface.
What to Remember
- Euclidean geometry describes flat space.
- Non-Euclidean geometries use different spatial rules.
- Euclidean space has zero curvature.
- Spherical geometry has positive curvature.
- Hyperbolic geometry has negative curvature.
- Geodesics generalize straight lines to curved spaces.
- Great circles are spherical geodesics.
- Euclidean triangles have angle sum \(180^\circ\).
- Spherical triangles have angle sum greater than \(180^\circ\).
- Hyperbolic triangles have angle sum less than \(180^\circ\).
- Parallel behavior differs across Euclidean, spherical, and hyperbolic geometry.
- Models can represent a geometry without visually resembling its intrinsic structure.
Where We Go Next
Euclidean and non-Euclidean spaces show that geometric relationships depend on the structure of the space in which they are defined. The next step is to examine how those spaces and objects are represented for reasoning, communication, and calculation.
The next lesson, Diagrams, Models, and Spatial Abstraction, develops the distinction between mathematical objects and the drawings, coordinate systems, models, projections, and abstractions used to represent them.