To denote geometric shapes and their projections, to display the relationship between them, as well as for the brevity of writing geometric sentences, algorithms for solving problems and proving theorems in the course uses geometric language, composed of the notation and symbols adopted in the mathematics course (in particular, in the new geometry course in high school).
All the variety of designations and symbols, as well as the connections between them, can be divided into two groups:
group I - designations of geometric figures and relationships between them;
group II designations of logical operations that form the syntactic basis of a geometric language.
Following is the full list mathematical symbols used in this course. Special attention given to symbols that are used to denote the projections of geometric shapes.
Group I
SYMBOLS DESIGNATING GEOMETRIC FIGURES AND THE RELATIONSHIP BETWEEN THEM
A. Designation of geometric shapes
1. The geometric figure is designated - F.
2. Points are indicated by capital letters Latin alphabet or Arabic numerals:
A, B, C, D, ..., L, M, N, ...
1,2,3,4,...,12,13,14,...
3. Lines arbitrarily located in relation to the projection planes are denoted by lowercase letters of the Latin alphabet:
a, b, c, d, ..., l, m, n, ...
Level lines are indicated by: h - horizontal; f - frontal.
The following designations are also used for straight lines:
(AB) - a straight line passing through points A and B;
[AB) - ray with origin at point A;
[AB] - a line segment bounded by points A and B.
4. Surfaces are designated by lowercase Greek letters:
α, β, γ, δ,...,ζ,η,ν,...
To emphasize the way of defining the surface, you should indicate the geometric elements that define it, for example:
α (a || b) - the plane α is determined by parallel straight lines a and b;
β (d 1 d 2 gα) - the surface β is determined by the guides d 1 and d 2, generating g and the plane of parallelism α.
5. Angles are indicated by:
∠ABC is the angle with the vertex at point B, as well as ∠α °, ∠β °, ..., ∠φ °, ...
6. Angular: value ( degree measure) is indicated by a sign that is placed above the corner:
Angle value ABC;
The value of the angle φ.
A right angle is marked with a square with a dot inside
7. Distances between geometric shapes are indicated by two vertical lines - ||.
For example:
| AB | - the distance between points A and B (the length of the segment AB);
| Aa | - distance from point A to line a;
| Аα | - distances from point A to surface α;
| ab | - the distance between lines a and b;
| αβ | distance between surfaces α and β.
8. For planes of projections, the designations are adopted: π 1 and π 2, where π 1 - horizontal plane projections;
π 2 -fryuntal plane of projections.
When replacing projection planes or introducing new planes, the latter denote π 3, π 4, etc.
9. The projection axes are designated: x, y, z, where x is the abscissa axis; y - ordinate axis; z - axis of the applicator.
The permanent straight line of the Monge plot is denoted by k.
10. Projections of points, lines, surfaces, any geometric figure are designated by the same letters (or numbers) as the original, with the addition of a superscript corresponding to the projection plane on which they are obtained:
A ", B", C ", D", ..., L ", M", N ", horizontal projections of points; A", B ", C", D ", ..., L", M " , N ", ... frontal projections of points; a ", b", c ", d", ..., l ", m", n ", - horizontal projections of lines; a", b ", c", d ", ..., l", m ", n", ... frontal projections of lines; α ", β", γ ", δ", ..., ζ ", η", ν ", ... horizontal projections of surfaces; α", β ", γ", δ ", ..., ζ" , η ", ν", ... frontal projections of surfaces.
11. Traces of planes (surfaces) are denoted by the same letters as the horizontal or frontal, with the addition of a subscript 0α, emphasizing that these lines lie in the projection plane and belong to the plane (surface) α.
So: h 0α - horizontal trace of the plane (surface) α;
f 0α - frontal trace of the plane (surface) α.
12. Traces of straight lines (lines) are denoted by capital letters, with which the words begin, which define the name (in Latin transcription) of the projection plane, which the line intersects, with a subscript indicating belonging to the line.
For example: H a - horizontal trace of a straight line (line) a;
F a - frontal trace of a straight line (line) a.
13. The sequence of dots, lines (of any figure) is marked with subscripts 1,2,3, ..., n:
A 1, A 2, A 3, ..., A n;
a 1, a 2, a 3, ..., a n;
α 1, α 2, α 3, ..., α n;
Ф 1, Ф 2, Ф 3, ..., Ф n, etc.
The auxiliary projection of the point, obtained as a result of the transformation to obtain the actual value of the geometric figure, is denoted by the same letter with the subscript 0:
A 0, B 0, C 0, D 0, ...
Axonometric projections
14. Axonometric projections of points, lines, surfaces are denoted by the same letters as the nature with the addition of a superscript 0:
A 0, B 0, C 0, D 0, ...
1 0 , 2 0 , 3 0 , 4 0 , ...
a 0, b 0, c 0, d 0, ...
α 0, β 0, γ 0, δ 0, ...
15. Secondary projections are indicated by adding superscript 1:
A 1 0, B 1 0, C 1 0, D 1 0, ...
1 1 0 , 2 1 0 , 3 1 0 , 4 1 0 , ...
a 1 0, b 1 0, c 1 0, d 1 0, ...
α 1 0, β 1 0, γ 1 0, δ 1 0, ...
To make it easier to read the drawings in the textbook, in the design of the illustrative material, several colors were used, each of which has a certain semantic meaning: the initial data are indicated by black lines (dots); green color used for lines of auxiliary graphic constructions; red lines (dots) show the results of constructions or those geometric elements that should be paid special attention to.
| No. | Designation | Content | An example of a symbolic notation |
|---|---|---|---|
| 1 | ≡ | Match | (AB) ≡ (CD) - a straight line passing through points A and B, coincides with the straight line passing through points C and D |
| 2 | ≅ | Congruent | ∠ABC≅∠MNK - angle ABC is congruent to angle MNK |
| 3 | ∼ | Are similar | ΔАВС∼ΔMNK - triangles ABC and MNK are similar |
| 4 | || | Parallel | α || β - plane α is parallel to plane β |
| 5 | ⊥ | Perpendicular | a⊥b - straight lines a and b are perpendicular |
| 6 | Interbreed | c d - straight lines c and d intersect | |
| 7 | Tangents | t l - line t is tangent to line l. βα - plane β tangent to the surface α |
|
| 8 | → | Displayed | Ф 1 → Ф 2 - the figure Ф 1 is displayed on the figure Ф 2 |
| 9 | S | Projection center. If the projection center is not its own point, then its position is indicated by an arrow, indicating the direction of projection | - |
| 10 | s | Projection direction | - |
| 11 | P | Parallel projection | p s α Parallel projection - Parallel projection on the plane α in the direction s |
| No. | Designation | Content | An example of a symbolic notation | An example of symbolic notation in geometry |
|---|---|---|---|---|
| 1 | M, N | The sets | - | - |
| 2 | A, B, C, ... | Elements of the set | - | - |
| 3 | { ... } | Comprises... | Ф (A, B, C, ...) | Ф (A, B, C, ...) - the figure Ф consists of points A, B, C, ... |
| 4 | ∅ | Empty set | L - ∅ - the set L is empty (contains no elements) | - |
| 5 | ∈ | Belongs to, is an element | 2∈N (where N is the set natural numbers) - number 2 belongs to set N | А ∈ а - point А belongs to line а (point A lies on line a) |
| 6 | ⊂ | Includes, contains | N⊂М - the set N is a part (subset) of the set M of all rational numbers | a ⊂α is a straight line a belongs to the plane α (understood in the sense of: the set of points of the straight line a is a subset of points of the plane α) |
| 7 | ∪ | Union | С = A U В - the set С is the union of the sets A and B; (1, 2.3, 4.5) = (1,2,3) ∪ (4,5) | ABCD = ∪ [ВС] ∪ - broken line, ABCD is union of segments [AB], [BC], |
| 8 | ∩ | Intersection of many | М = К∩L - the set М is the intersection of the sets К and L (contains elements belonging to both the set K and the set L). M ∩ N = ∅ - the intersection of the sets M and N is an empty set (the sets M and N have no common elements) | a = α ∩ β is a straight line a is an intersection planes α and β a ∩ b = ∅ - lines a and b do not intersect (do not have common points) |
| No. | Designation | Content | An example of a symbolic notation |
|---|---|---|---|
| 1 | ∧ | Conjunction of sentences; matches the union "and". Sentence (p∧q) is true if and only if p and q are both true | α∩β = (K: K∈α∧K∈β) The intersection of surfaces α and β is a set of points (line), consisting of all those and only those points K that belong to both the surface α and the surface β |
| 2 | ∨ | Disjunction of sentences; matches the conjunction "or". Sentence (p∨q) is true when at least one of the sentences p or q (that is, either p, or q, or both) is true. | - |
| 3 | ⇒ | Implication is a logical consequence. The sentence p⇒q means: "if p, then q" | (a || c∧b || c) ⇒a || b. If two straight lines are parallel to the third, then they are parallel to each other |
| 4 | ⇔ | The sentence (p⇔q) is understood in the sense: "if p, then q; if q, then p" | А∈α⇔А∈l⊂α. A point belongs to a plane if it belongs to some line belonging to this plane. The converse is also true: if a point belongs to some line, belonging to the plane, then it belongs to the plane itself |
| 5 | ∀ | The quantifier of generality is read: for everyone, for everyone, for everyone. The expression ∀ (x) P (x) means: "for any x: the property P (x) holds" | ∀ (ΔABS) (= 180 °) For any (for any) triangle, the sum of the values of its angles at the vertices is 180 ° |
| 6 | ∃ | The quantifier of existence, reads: exists. The expression ∃ (x) P (x) means: "there is an x with the property P (x)" | (∀α) (∃a) For any plane α, there is a straight line a that does not belong to the plane α and parallel to the plane α |
| 7 | ∃1 | The quantifier of the uniqueness of existence, reads: there is only one (-th, -th) ... The expression ∃1 (x) (Px) means: "there is only one (only one) x, Px " | (∀ A, B) (A ≠ B) (∃1a) (a∋A, B) For any two different points A and B, there is a unique line a, passing through these points. |
| 8 | (Px) | Denial of the statement P (x) | ab (∃α) (α⊃a, b) If the lines a and b intersect, then there is no plane a that contains them |
| 9 | \ | Negation of a sign | ≠ -segment [AB] is not equal to the segment. A? B - line a is not parallel to line b |