geometric congruence

geometric congruence

Two geometrical constructs are

congruent

if there is a

finite sequence

of geometric transformations

mapping

each one into the other. In this entry, we discuss three types of

geometric congruences

:

congruence

(the usual congruence), affine congruence, and projective congruence. After discussing congruence, we will briefly discuss congruence in

Non-Euclidean geometry

before moving on to affine congruence.

Euclidean Congruence

In the usual

Euclidean space

, these

rigid motions

are

translations

,

rotations

, and

reflections

(and of course

compositions

of them).

In a less formal sense, saying two constructs are congruent amounts to saying that the two constructs are essentially “the same” under the

geometry

that is being used.

The following are criteria that indicate that two given

triangles

are congruent:

•

SSS. If two triangles have their corresponding sides equal, they are congruent.

•

SAS. If two triangles have two corresponding sides equal as well as the angle between them, the triangles are congruent.

•

ASA. If two triangles have 2 pairs of corresponding angles equal, as well as the side between them, the triangles are congruent.

•

AAS. (Also known as SAA.) If two triangles have 2 pairs of corresponding angles equal, as well as a pair of corresponding sides which is not in between them, the triangles are congruent.

Congruence in Non-Euclidean Geometry

Note that the criteria listed above are also valid in hyperbolic geometry (and therefore in

neutral geometry

). Also note that

AAS is not valid in spherical geometry

, but all of the other criteria are. On the other hand, in both hyperbolic geometry and

spherical geometry

, AAA is a criterion that indicates that two given triangles are congruent.

Affine Congruence

Two geometric figures in an

affine space

are

affine congruent

if there is an

affine transformation

mapping one figure to another.
Since lengths and angles are not preserved by affine transformations, the class of specific geometric

configurations

is wider
than that of the class of the same geometric configuration under Euclidean congruence. For example, all triangles are
affine congruent, whereas Euclidean congruent triangles are confined only to those that are SSS, SAS, or ASA. Another
example is found in the class of

ellipses

, which contains ellipses of all sizes and shapes, including circles. However, in
Euclidean congruence, circles are only congruent to circles of the same radius.

Projective Congruence

Two geometric figures in a

projective space

are

projective congruent

if there is a

projective transformation

mapping one
figure to another. Here, we find the class of congruent objects of a geometric shape even wider than the class of
congruent objects of the same geometric shape under affine congruence. For example, the class of all

conic sections

are
projectively congruent, so a circle is projectively the same as an ellipse, as a

parabola

, and as a

hyperbola

. Of course,
under affine congruence, parabola, hyperbola, and ellipse are three distinct geometric objects.

Title

geometric congruence

Canonical

name

GeometricCongruence

Date of creation

2013-03-22 12:11:21

Last modified on

2013-03-22 12:11:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

22

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51M99

Synonym

Euclidean congruent

Synonym

affine congruent

Synonym

projective congruent

Synonym

congruent

Related topic

TriangleSolving

Related topic

UsingPrimitiveRootsAndIndexToSolveCongruences

Related topic

NormalOfPlane

Related topic

CenterNormalAndCenterNormalPlaneAsLoci

Defines

SSS

Defines

SAS

Defines

ASA

Defines

SAA

Defines

AAS

Defines

Euclidean congruence

Defines

affine congruence

Defines

projective congruence

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