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Basic concepts
Objects of space and their mutual position
- The concepts of point, line, plane and incidency are basic concepts, they are not defined. (Figure 1)
- Two different lines are intersecting if they have a common point. (Figure 2/a)
- Two lines are parallel if they are in the same plane and they do not share a common point; in addition, every line is parallel to itself. (Figure 2/b)
- Two lines are skew if they are not contained in one plane. (Figure 2/c)
\latex{ R }
\latex{ Q }
\latex{ P }
\latex{ S }
\latex{ e }
Figure 1
\latex{e\cap f=M}
\latex{e\cap f=\varnothing (e||f)}
\latex{ M }
\latex{ f }
\latex{ f }
\latex{ f }
\latex{ e }
\latex{ e }
\latex{ e }
\latex{ a })
\latex{ b })
\latex{ c })
Figure 2
- Two different planes are intersecting if they have exactly one line in common. (Figure 3/a)
- Two planes are parallel if they does not have any points in common; in addition, every plane is parallel to itself.
- A line is incident to a plane if any point of the line is also a point of the plane. (Figure 3/b)
- A line intersects a plane if they have exactly one point in common.
- A line and a plane are parallel if they do not have any points in common.
\latex{S_{1}\cap S_{2}=e}
\latex{e\subset S}
\latex{g\cap S=\varnothing (g||S)}
\latex{f\cap S=M}
\latex{ e }
\latex{ S }
\latex{ M }
\latex{ f }
\latex{ g }
\latex{ S_2 }
\latex{ S_1 }
\latex{ e }
\latex{ a })
\latex{ b })
Figure 3
Ray, line segment, half-plane, half-space, angle
- A line is divided into two rays by any of its points.
- Two points of a line define a line segment.
- A plane is divided into two half-planes by any of its lines. (Figure 4)
\latex{ e }
Figure 4
- The space is divided into two half-spaces by any of its planes. (Figure 5)
- Any two rays starting at the same point divide the plane into two parts. The name of one such part is angle. (Figure 6/a)
- By rotating a ray around its starting point in any direction we get a rotation angle (Figure 6/b).
The rotation angle is positive if the direction of the rotation is counter-clockwise and it is negative if the direction is clockwise. (Figure 6/c).
\latex{ S }
Figure 5
angle
side
side
vertex
\latex{\alpha}
\latex{\alpha}
\latex{\alpha \gt 0}
rotation in positive
direction
direction
counter-clockwise
\latex{\alpha}
\latex{\beta}
\latex{\beta \lt 0}
rotation in negative
direction
direction
clockwise
\latex{ O }
\latex{ O }
\latex{ O }
\latex{ O }
\latex{ a })
\latex{ b })
\latex{ c })
Figure 6
Types of angles (Figure 7)
\latex{\alpha}
\latex{0°\lt \alpha \lt 90°}
actual angle
\latex{\alpha}
\latex{\alpha =90°}
right angle
\latex{\alpha}
\latex{90°\lt \alpha \lt 180°}
obtuse angle
\latex{\alpha =180°}
\latex{\alpha}
straight angle
\latex{\alpha}
reflex (concave) angle
\latex{180°\lt \alpha \lt 360°}
\latex{\alpha}
full angle
\latex{\alpha =360°}
convex angles
\latex{0°\lt \alpha \lt 180°}
Figure 7
Notable pairs of angles
- If two angles share a common vertex and their sides can be paired up such that the pairs are each other's extension then they are called vertical angles. Vertical angles are of equal size. (Figure 8)
- If the two angles have one arm in common while the other two are two rays with a common starting point forming a line together, then they are called a linear pair of angles. The sum of a pair of linear angles is \latex{ 180º }. (Figure 8)
- If the sum of two angles is \latex{ 180º } then they are called supplementary angles.
- If the sum of two angles is \latex{ 90º } then they are called complementary angles.
- If the arms of either two convex or two concave angles are pairwise pointing the same direction, then they are called corresponding angles, while if they are pairwise pointing the opposite direction then they are called alternate angles. (Figure 9)
\latex{\alpha =\beta ,}
corresponding
angles
angles
\latex{\alpha}
\latex{\beta}
\latex{\gamma}
\latex{\delta}
\latex{\gamma =\delta ,}
alternate angles
Figure 9
\latex{\alpha}
\latex{\beta}
\latex{\alpha '}
\latex{\alpha =\alpha ';\alpha +\beta =180°}
Figure 8
- If both of the angles with mutually perpendicular arms are acute angles or both are obtuse angles, then they are of equal size; if one of them is an acute angle, the other one is an obtuse angle, then these are supplementary angles. (Figure 10)
\latex{\alpha}
\latex{\beta}
\latex{\alpha =\beta}
\latex{\gamma}
\latex{\delta}
\latex{\delta =\gamma}
\latex{\alpha}
\latex{\beta}
\latex{\alpha+\beta =180°}
Figure 10
Distance of objects in space
- The distance of two points is the length of the line segment joining them.
- The distance of a point and a straight line is the distance of the point and the base point of the line segment through the point and perpendicular to the straight line. If the point is incident to the line then their distance is \latex{ 0 }.
\latex{d(e;f)=d(P;f)=d(Q;e)=PQ}
\latex{ P }
\latex{ Q }
\latex{ f }
\latex{ e }
Figure 11
- The distance of two parallel lines is the distance of one line and an arbitrary point from the other line. (Figure 11)
- The distance of intersecting lines is \latex{ 0 }.
- The distance of a point and a plane is the length of the line segment between the point and the plane which is perpendicular to the plane. (Figure 12)
\latex{d(P;S)=d(P;T)=PT}
\latex{ P }
\latex{ T }
\latex{ S }
Figure 12
*DEFINITION: A straight line is perpendicular to a plane if it is perpendicular to every line of the plane.
*THEOREM: If a straight line is perpendicular to two different straight lines from a plane which are incident to the common point of the original line and the plane, then the line is perpendicular to the plane.
- The distance of two parallel planes is the distance of one plane and an arbitrary point from the other plane. (Figure 13)
- The distance of two intersecting planes is \latex{ 0 }.
- The distance of two skew lines is the distance of the two parallel planes both incident to one of the lines. (It is possible to prove that for a given pair of skew lines these planes are uniquely determined.)
\latex{d(S_{1};S_{2} )=d(P;S_{2} )=d(Q;S_{1} )=PQ}
\latex{ P }
\latex{ Q }
\latex{ S_2 }
\latex{ S_1 }
Figure 13
Angle of objects in space
- Two intersecting straight lines in space determine four angles, from which the opposite pairs are identical (vertical angles). The smaller (not greater) of the two different neighbouring angles is the angle between the two lines (Figure 14). The angle between parallel lines is \latex{ 0º }.
- If a line is not perpendicular to a plane then the angle between the line and the plane is the angle between the line and its orthogonal projection to the plane. (Figure 15)
Figure 14
*DEFINITION: The orthogonal projection of a point to a plane is the base point of the perpendicular to the plane through the point. The orthogonal projection of a shape to a plane is the set of the orthogonal projections of its points.
\latex{\alpha}
\latex{ e }
\latex{ S }
Figure 15
*THEOREM: If a line is not perpendicular to a plane then its orthogonal projection to the plane is a line.
- Take an arbitrary point of the line of intersection of two intersecting planes. Take the straight line perpendicular to the line of intersection in both planes. The dihedral angle of the two planes is the angle between these two perpendicular lines. (Figure 16)
\latex{\alpha}
Figure 16
Notable point sets
I. The perpendicular bisector
DEFINITION: The perpendicular bisector of a line segment on the plane is the line perpendicular to the segment and incident to the midpoint of the segment.
\latex{ P }
\latex{ F }
\latex{ B }
\latex{ A }
\latex{ Q }
Figure 17
THEOREM: The set of points equidistant from the points \latex{ A } and \latex{ B } in the plane form the perpendicular bisector of the segment \latex{ AB }. (Figure 17)
*THEOREM: The set of points equidistant from points \latex{ A } and \latex{ B } in the space form the bisecting plane, the plane perpendicular to the segment \latex{ AB } and incident to the midpoint of it. (Figure 18)
\latex{ A }
\latex{ F }
\latex{ B }
Figure 18
II. The angle bisector
DEFINITION: The bisector of a convex angle is the ray with endpoint being the vertex of the angle contained in the angle such that it divides the angle into two congruent (equal) angles.
THEOREM: In a convex angle the set of points equidistant from the two arms form the bisector of the angle. (Figure 19)
\latex{ C }
\latex{ M }
\latex{ d }
\latex{ d }
Figure 19
The angle bisector can be constructed following the steps on Figure 20.
\latex{CA=CB}
\latex{AM=BM}
\latex{ A }
\latex{ A }
\latex{ A }
\latex{ M }
\latex{ M }
\latex{ C }
\latex{ C }
\latex{ C }
\latex{ B }
\latex{ B }
\latex{ B }
\latex{ 3. }
\latex{ 2. }
\latex{ 1. }
Figure 20
III. The circle
DEFINITION: The set of points which are at a given distance \latex{ r } from a given point \latex{ O } form a circle. (Figure 21)
*DEFINITION: The set of points which are at a distance not greater than a given distance \latex{ r } from a given point \latex{ O } form a closed disc with centre \latex{ O } and radius \latex{ r }. (Figure 22)
*DEFINITION: The set of points which are at a distance less than the given distance \latex{ r } from a given point \latex{ O } form an open disc with centre \latex{ O } and radius \latex{ r }. (Figure 23)
\latex{OP\lt r}
OPEN DISC
\latex{ O }
\latex{ P }
\latex{ r }
Figure 23
\latex{OP\leq r}
CLOSED DISC
\latex{ O }
\latex{ r }
\latex{ P }
Figure 22
\latex{ r }: radius
\latex{ O }: center
\latex{OP=r}
CIRCLE
\latex{ P }
Figure 21
Notations concerning the circle, disc and their parts can be seen on Figure 24.
CHORD
DIAMETER
SECANT LINE
SEGMENT
SECTOR
ARC
concentric circles (circles with common centre)
RING
or
ANNULUS
Figure 24
DEFINITION: A tangent of a circle is a line in the plane of the circle which has exactly one point in common with the circle. (Figure 25)
point
of tangency
of tangency
TANGENT
LINE
LINE
\latex{ O }
\latex{ r }
Figure 25
THEOREM: The tangent of a circle is perpendicular to the radius at the point of tangency.
The tangent at a given point \latex{ P } of a given circle can be constructed following the steps on Figure 26.
\latex{KP=PL}
\latex{KM=ML}
\latex{ O }
\latex{ P }
\latex{ O }
\latex{ K }
\latex{ P }
\latex{ L }
\latex{ O }
\latex{ K }
\latex{ P }
\latex{ L }
\latex{ O }
\latex{ K }
\latex{ P }
\latex{ L }
\latex{ M }
\latex{ M }
\latex{ e }
\latex{ 4. }
\latex{ 3. }
\latex{ 2. }
\latex{ 1. }
Figure 26
IV. The sphere
DEFINITION: The set of points which are at a given distance from a given point is called a sphere.
V. The conic sections
THE PARABOLA
DEFINITION: The parabola is the set of points on the plane equidistant from a given straight line d of the plane and a point \latex{ F } not lying on \latex{ d }. (Figure 27)
\latex{ F }
\latex{ T }
\latex{ p }
\latex{ d }
\latex{ P }
\latex{ t }
Figure 27
\latex{ F } is the focus (focal point) of the parabola, \latex{ d } is its directrix.
THE ELLIPSE
*DEFINITION:The ellipse is the set of points on the plane for which the sum of the distances from two given points of the plane is a given constant (greater than the distance of the two given points). (Figure 28)
The two given points are called the foci (plural or focus) or focal points of the ellipse.
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ O }
\latex{ P }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ a }
\latex{ a }
\latex{ c }
\latex{ c }
\latex{ F_1 }
\latex{ F_2 }
\latex{ r_2 }
\latex{ r_1 }
\latex{ a }
Figure 28
THE HYPERBOLA
*DEFINITION:The hyperbola is the set of points on the plane for which the absolute value of the difference between the distances from two given points of the plane is a given constant (smaller than the distance of the two given points). (Figure 29)
\latex{ O }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ c }
\latex{ c }
\latex{ a }
\latex{ a }
\latex{ F_1 }
\latex{ F_2 }
\latex{ r_2 }
\latex{ P }
\latex{ r_1 }
\latex{ b }
\latex{ b }
Figure 29
The two given points are called the foci (plural or focus) or focal points of the hyperbola.
Exercises
{{exercise_number}}. The distance between points \latex{ A } and \latex{ B } is \latex{ 4\, cm }. Decide which of the following statements are true and which are false. If, for a point \latex{ P } of the plane,
- \latex{PA\lt 2\,cm}, then \latex{PB\lt 5\,cm};
- \latex{PA\lt 1\,cm}, then \latex{PB\gt 2\,cm}.
{{exercise_number}}. What should the distance between points \latex{ A } and \latex{ B } be if the following statements are true? If, for a point \latex{ P } of the plane,
- \latex{PA\lt 3\,cm}, then \latex{PB\lt 7\,cm};
- \latex{PA\lt 3\,cm}, then \latex{PB\gt 7\,cm}.
{{exercise_number}}. Five angles together form a complete angle. It is also known that every angle is \latex{ 15º } greater than the preceding one. Determine the size of each angle.
{{exercise_number}}. A ship leaves the port in a northern direction. After two days it turns by \latex{ –45º }, then after another two days it turns by \latex{ +150º }. Compared to the original (northern) direction, which direction is it moving now?
{{exercise_number}}. The shape of a park is a rectangle which is not a square. Following the inner angle bisectors there are four footpaths starting from the corners of the park and ending on the boundary as well. What is the shape determined by the four footpaths in the middle of the park?
{{exercise_number}}. Take a pair of intersecting straight lines and a circle on the plane. Construct those points of the circle which are equidistant from the two lines. How many such points can possibly exist? Does there always exist such a point?
{{exercise_number}}. Determine the number of intersection points of ten straight lines on the plane if exactly three of them are parallel and exactly three of them intersect in one point.
{{exercise_number}}. Determine the maximum number of regions of the space if it is divided by
- three planes;
- four planes;
- a sphere and three planes;
- a sphere and four planes.