The course uses geometric language, composed of notations and symbols adopted in a mathematics course (in particular, in the new geometry course in high school).
The whole 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 the geometric language.
Below is full list mathematical symbols used in this course. Special attention devoted to symbols that are used to designate projections of geometric figures.
Group I
SYMBOLS INDICATING GEOMETRIC FIGURES AND RELATIONS BETWEEN THEM
A. Designation of geometric figures
1. A geometric figure is designated - F.
2. Points are designated in 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 designated by lowercase letters of the Latin alphabet:
a, b, c, d, ... , l, m, n, ...
Level lines are designated: h - horizontal; f- front.
The following notations are also used for straight lines:
(AB) - a straight line passing through points A and B;
[AB) - ray with beginning at point A;
[AB] - a straight line segment bounded by points A and B.
4. Surfaces are designated by lowercase letters of the Greek alphabet:
α, β, γ, δ,...,ζ,η,ν,...
To emphasize the way a surface is defined, the geometric elements by which it is defined should be indicated, for example:
α(a || b) - the plane α is determined by parallel lines a and b;
β(d 1 d 2 gα) - the surface β is determined by the guides d 1 and d 2, the generator g and the plane of parallelism α.
5. Angles are indicated:
∠ABC - angle with vertex at point B, as well as ∠α°, ∠β°, ... , ∠φ°, ...
6. Angular: value ( degree measure) is indicated by the sign placed above the angle:
The magnitude of the angle ABC;
The magnitude of the angle φ.
A right angle is marked with a square with a dot inside
7. The distances between geometric figures are indicated by two vertical segments - ||.
For example:
|AB| - the distance between points A and B (length of segment AB);
|Aa| - distance from point A to line a;
|Aα| - distances from point A to surface α;
|ab| - distance between lines a and b;
|αβ| distance between surfaces α and β.
8. For projection planes, the following designations are accepted: π 1 and π 2, where π 1 is the horizontal projection plane;
π 2 - frontal projection plane.
When replacing projection planes or introducing new planes, the latter are designated π 3, π 4, etc.
9. The projection axes are designated: x, y, z, where x is the abscissa axis; y - ordinate axis; z - applicate axis.
Monge's constant straight line diagram is denoted by k.
10. Projections of points, lines, surfaces, any geometric figure are indicated by the same letters (or numbers) as the original, with the addition of a superscript corresponding to the projection plane on which they were 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 designated by the same letters as horizontal or frontal, with the addition of the 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 indicated by capital letters, with which the words begin that define the name (in Latin transcription) of the projection plane that the line intersects, with a subscript indicating the affiliation with the line.
For example: H a - horizontal trace of a straight line (line) a;
F a - frontal trace of straight line (line) a.
13. The sequence of points, lines (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 a point, obtained as a result of transformation to obtain the actual value of a geometric figure, is denoted by the same letter with a 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 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 a 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, several colors are used when designing the illustrative material, each of which has a certain semantic meaning: black lines (dots) indicate the original data; green color used for lines of auxiliary graphic constructions; red lines (dots) show the results of constructions or those geometric elements to which special attention should be paid.
| No. by por. | Designation | Content | Example of symbolic notation |
|---|---|---|---|
| 1 | ≡ | Match | (AB)≡(CD) - a straight line passing through points A and B, coincides with the line passing through points C and D |
| 2 | ≅ | Congruent | ∠ABC≅∠MNK - angle ABC is congruent to angle MNK |
| 3 | ∼ | Similar | ΔАВС∼ΔMNK - triangles АВС 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 | Crossbreed | c d - straight lines c and d intersect | |
| 7 | Tangents | t l - line t is tangent to line l. βα - plane β tangent to surface α |
|
| 8 | → | Displayed | F 1 →F 2 - figure F 1 is mapped to figure F 2 |
| 9 | S | Projection Center. If the projection center is an improper point, then its position is indicated by an arrow, indicating the direction of projection | - |
| 10 | s | Projection direction | - |
| 11 | P | Parallel projection | р s α Parallel projection - parallel projection onto the α plane in the s direction |
| No. by por. | Designation | Content | Example of symbolic notation | Example of symbolic notation in geometry |
|---|---|---|---|---|
| 1 | M,N | Sets | - | - |
| 2 | A,B,C,... | Elements of the set | - | - |
| 3 | { ... } | Comprises... | Ф(A, B, C,...) | Ф(A, B, C,...) - figure Ф consists of points A, B, C, ... |
| 4 | ∅ | Empty set | L - ∅ - set L is empty (does not contain elements) | - |
| 5 | ∈ | Belongs to, is an element | 2∈N (where N is the set natural numbers) - the number 2 belongs to the set N | A ∈ a - point A belongs to line a (point A lies on line a) |
| 6 | ⊂ | Includes, contains | N⊂M - set N is part (subset) of set M of all rational numbers | a⊂α - straight line a belongs to the plane α (understood in the sense: the set of points of the line a is a subset of the points of the plane α) |
| 7 | ∪ | An association | C = A U B - set C is a union of sets A and B; (1, 2. 3, 4.5) = (1,2,3)∪(4.5) | ABCD = ∪ [ВС] ∪ - broken line, ABCD is combining segments [AB], [BC], |
| 8 | ∩ | Intersection of many | M=K∩L - the set M is the intersection of the sets K 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 the empty set (sets M and N do not have common elements) | a = α ∩ β - straight line a is the intersection planes α and β a ∩ b = ∅ - straight lines a and b do not intersect (Dont Have common points) |
| No. by por. | Designation | Content | Example of symbolic notation |
|---|---|---|---|
| 1 | ∧ | Conjunction of sentences; corresponds to the conjunction "and". A sentence (p∧q) is true if and only if p and q are both true | α∩β = (К: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) true when at least one of the sentences p or q is true (that is, either p or q, or both). | - |
| 3 | ⇒ | Implication is a logical consequence. The sentence p⇒q means: “if p, then q” | (a||c∧b||c)⇒a||b. If two lines are parallel to a third, then they are parallel to each other |
| 4 | ⇔ | The sentence (p⇔q) is understood in the sense: “if p, then also q; if q, then also p” | А∈α⇔А∈l⊂α. A point belongs to a plane if it belongs to some line belonging to this plane. The converse statement is also true: if a point belongs to a certain line, belonging to the plane, then it belongs to the plane itself |
| 5 | ∀ | The general quantifier reads: for everyone, for everyone, for anyone. The expression ∀(x)P(x) means: “for every x: the property P(x) holds” | ∀(ΔАВС)( = 180°) For any (for any) triangle, the sum of the values of its angles at vertices equals 180° |
| 6 | ∃ | The existential quantifier reads: exists. The expression ∃(x)P(x) means: “there is an x that has 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 (-i, -th)... The expression ∃1(x)(Рх) means: “there is only one (only one) x, having the property Px" | (∀ A, B)(A≠B)(∃1a)(a∋A, B) For any two various points A and B there is a single straight line a, passing through these points. |
| 8 | (Px) | Negation of the statement P(x) | ab(∃α)(α⊃a, b).If lines a and b intersect, then there is no plane a that contains them |
| 9 | \ | Negation of the sign | ≠ -segment [AB] is not equal to segment .a?b - line a is not parallel to line b |
A point and a straight line are the basic geometric figures on a plane.
The ancient Greek scientist Euclid said: “a point” is something that has no parts.” The word "point" translated from Latin language means the result of an instant touch, a prick. A point is the basis for constructing any geometric figure.
A straight line or simply a straight line is a line along which the distance between two points is the shortest. A straight line is infinite, and it is impossible to depict the entire straight line and measure it.
Points are denoted by capital Latin letters A, B, C, D, E, etc., and straight lines by the same letters, but lowercase a, b, c, d, e, etc. A straight line can also be denoted by two letters corresponding to points lying on her. For example, straight line a can be designated AB.
We can say that points AB lie on line a or belong to line a. And we can say that straight line a passes through points A and B.
Protozoa geometric figures on a plane it is a segment, a ray, a broken line.
A segment is a part of a line that consists of all points of this line, limited by two selected points. These points are the ends of the segment. A segment is indicated by indicating its ends.
A ray or half-line is a part of a line that consists of all points of this line lying on one side of a given point. This point is called the starting point of the half-line or the beginning of the ray. The beam has a starting point, but no end.
Half-lines or rays are designated by two lowercase Latin letters: the initial and any other letter corresponding to a point belonging to the half-line. In this case, the starting point is placed in the first place.
It turns out that the straight line is infinite: it has neither beginning nor end; a ray has only a beginning, but no end, but a segment has a beginning and an end. Therefore, we can only measure a segment.
Several segments that are sequentially connected to each other so that the segments (neighboring) that have one common point are not located on the same straight line represent a broken line.
A broken line can be closed or open. If the end of the last segment coincides with the beginning of the first, we have a closed broken line; if not, it is an open line.
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Despite the fact that geometry is one of the exact sciences, scientists cannot unambiguously define the term “straight line”. In the very general view we can give the following definition: “A straight line is a line along which the path is equal to the distance between two points.”
What is a straight line in mathematics? The definition of a straight line in mathematics is that a straight line has no ends and can continue in both directions indefinitely.
The basic concepts of geometry include point, line and plane; they are given without definition, but definitions of other geometric figures are given through these concepts. A plane, like a straight line, is a primary concept that has no definition. This statement is established by the following axiom: if two points of a line lie in a certain plane, then all points of this line lie in this plane. And the statement itself that is being proven is called a theorem. The formulation of the theorem usually consists of two parts.
Problem: where is the line, ray, segment, curve? The vertices of a broken line (similar to the tops of mountains) are the point from which the broken line begins, the points at which the segments that form the broken line are connected, the point at which the broken line ends. Problem: which broken line is longer, and which more peaks? Adjacent sides of a polygon are adjacent links of a broken line. The vertices of a polygon are the vertices of a broken line. Adjacent vertices are the endpoints of one side of the polygon.
In mathematics lessons you can hear the following explanation: a mathematical segment has a length and ends. A segment in mathematics is the set of all points lying on a straight line between the ends of the segment.
In the future there will be definitions for different figures except two - a point and a straight line. This means that sometimes we can denote a straight line with two capital Latin letters, for example, straight line \(AB\), since no other straight line can be drawn through these two points. Symbolically we write the segment \(AB\).
What is a point in mathematics?
Theorem: The midline of a triangle is parallel to one of its sides and equal to half of that side. C. Altitude of a right triangle drawn from the vertex right angle, divides a triangle into two similar ones right triangle, each of which is similar to a given triangle. C. An inscribed angle subtended by a semicircle is a right angle. Here are the basic definitions, theorems, and properties of figures on the plane.
The vector with the coordinates of the point is called a normal vector; it is perpendicular to the line.
In a systematic presentation of geometry, a straight line is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry.
4. Two divergent lines on a plane either intersect at a single point, or they are parallel. A ray is a part of a straight line limited on one side. A segment, like a straight line, is denoted by either one letter or two. In the latter case, these letters indicate the ends of the segment.