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Geometric transformations
DEFINITION: A geometric transformation is a function (unambiguous assignment) which maps points to points, that is, its domain and codomain are point sets. (In a broader sense.)
DEFINITION: A geometric transformation is a bijective function whose domain is the set of points of the plane or the space and its codomain matches its domain. (In a narrower sense.)
CONGRUENT TRANSFORMATIONS
DEFINITION: The geometric transformations, in the case of which the image of any line segment is a line segment whose length is equal to the original length, are called congruent (distance-preserving) transformations. Another term for this type of transformations is isometry.
Congruent transformations in the plane
I. Reflection across a straight line
DEFINITION: Let \latex{t} be a straight line in the plane. To every point \latex{P} of the plane assign a point \latex{P’} such that
– if \latex{P \in t} then \latex{P’ = P};
– if \latex{P \notin t} then \latex{P’} is the point of the plane for which \latex{t} is the perpendicular bisector of the line segment \latex{PP’}.
– if \latex{P \notin t} then \latex{P’} is the point of the plane for which \latex{t} is the perpendicular bisector of the line segment \latex{PP’}.
\latex{t} is called the \latex{axis} of the reflection.
A reflection is uniquely determined by its axis \latex{t} or by a point \latex{P} and its image \latex{P}’ provided \latex{(P’\neq P)}. The image of a point \latex{P} can be constructed as seen on Figure 30.
\latex{ P }
\latex{ P }
\latex{ T }
\latex{ P' }
\latex{ T }
\latex{ t }
\latex{ t }
Figure 30
PROPERTIES OF REFLECTION ACROSS A LINE
- The points on the axis are fixed points and there are no other fixed points. (Figure 31)
- The straight lines perpendicular to the axis are invariant lines. (Figure 32)
- The image of any line which intersects the axis obliquely (not perpendicularly) intersects the original line on the axis and it includes the same angle with the axis as the original straight line. (Figure 33)
\latex{ P=P' }
\latex{ t }
Figure 31
- The image of a line parallel to the axis is parallel to the axis as well, and the axis bisects the region between the line and the image. (Figure 34)
\latex{ P }
\latex{ Q }
\latex{ Q' }
\latex{ P' }
\latex{ e=e' }
\latex{ t }
Figure 32
\latex{\alpha}
\latex{\alpha}
\latex{ e }
\latex{ t }
\latex{ e' }
Figure 33
\latex{ e }
\latex{ d }
\latex{ d }
\latex{ e' }
\latex{ t }
Figure 34
- Reflection across any line is a distance- and angle-preserving transformation, that is, the length of any segment and its image has the same length, and any angle and its image are congruent. (Figure 35)
- The transformation obtained as the result of two consecutive reflections through the same axis is the identical transformation (which maps every point to itself) or identity.
- The reflection across an arbitrary straight line is an orientation reversing geometric transformation.
\latex{\alpha}
\latex{\alpha}
\latex{PQ=P’Q’}
\latex{\alpha= \alpha’}
\latex{ Q }
\latex{ P }
\latex{ P' }
\latex{ Q' }
\latex{ t }
Figure 35
AXIALLY SYMMETRIC FIGURES
DEFINITION: An object in the plane is said to be axially symmetric if there exists a straight line in the plane for which the object is invariant for the reflection across said line.
II. Point reflection in the plane
DEFINITION: Let a point \latex{O} of the plane be given. To every point \latex{P} of the plane a point \latex{P’} is assigned the following way:
– the image of \latex{O} is itself, \latex{O’ = O};
– if \latex{P\neq O} then \latex{P’} is the point of \latex{P} for which \latex{O} is the midpoint of the line segment \latex{PP’}.
– if \latex{P\neq O} then \latex{P’} is the point of \latex{P} for which \latex{O} is the midpoint of the line segment \latex{PP’}.
\latex{O} called the centre of the reflection.
A reflection through a point is uniquely determined by the point \latex{O}, or a point \latex{P} different from \latex{O} and its image \latex{P’}.
PROPERTIES OF THE REFLECTION THROUGH A POINT
- The only fixed point of the transformation is the centre \latex{O} of the reflection.
- The invariant lines of the reflections are exactly the lines containing \latex{O}. (Figure 36)
- The image of a straight line not containing \latex{O} is the straight line parallel to the original line for which the distance of \latex{O} is equal from the two lines. (Figure 37)
\latex{ P }
\latex{ O }
\latex{ P' }
Figure 36
- Any reflection through a point is a distance- and angle-preserving transformation. (Figure 38)
- If, for a reflection through a given point, the image of a point \latex{P} is the point \latex{P’}, then the image of \latex{P’} for the same reflection is \latex{P}, that is, by repeating the same reflection twice, we get the identical transformation.
- Any reflection through a point can be produced by reflecting consecutively across two perpendicular lines. The centre of the reflection is the intersection of the two axes. (Figure 39)
- Reflection through any point is an orientation-preserving transformation.
\latex{ O }
\latex{ P }
\latex{ Q' }
\latex{ P' }
\latex{ Q }
\latex{ e }
\latex{ e' }
Figure 37
\latex{ C }
\latex{ C' }
\latex{ C'' }
\latex{ B' }
\latex{ B }
\latex{ A }
\latex{ A' }
\latex{ A'' }
\latex{ B'' }
\latex{ O }
\latex{ t_1 }
\latex{ t_2 }
Figure 39
\latex{AB=A’ B’}
\latex{B}
\latex{\alpha=\alpha’}
\latex{\alpha}
\latex{\alpha’}
\latex{ O }
\latex{ O }
\latex{ A }
\latex{ B' }
\latex{ A' }
Figure 38
CENTRALLY SYMMETRIC FIGURES
DEFINITION: A planar (or spatial) figure is centrally symmetric (point-symmetric) if there is a point in the plane (space) the figure reflected in which is an invariant figure.
III. Rotation about a point
DEFINITION: A point \latex{O} of the plane and a directed angle \latex{\alpha} is given. Assign a point \latex{P’} to each point \latex{P} in the plane as follows:
– the image of \latex{O} is itself, that is, \latex{O’ =O};
– if \latex{P\neq O}, then \latex{P’} is the point in the plane for which \latex{OP =OP’} and ray \latex{OP’} is the rotated image of ray \latex{OP} when rotated through directed rotation angle \latex{\alpha}.
– if \latex{P\neq O}, then \latex{P’} is the point in the plane for which \latex{OP =OP’} and ray \latex{OP’} is the rotated image of ray \latex{OP} when rotated through directed rotation angle \latex{\alpha}.
\latex{O} is called the centre of rotation. (Figure 40)
\latex{\color{de0454}\alpha \lt 0}
rotation
in negative
direction
in negative
direction
rotation
in positive
direction
in positive
direction
\latex{\color{2c7dc6}\alpha \gt 0}
\latex{\alpha}
\latex{ O }
\latex{ Q }
\latex{ P' }
\latex{ P }
\latex{ Q' }
Figure 40
Any rotation about some point is uniquely determined be the point \latex{O} and the directed angle \latex{\alpha} or the point \latex{O}, a point \latex{P} different from \latex{O} and its image \latex{P’}.
PROPERTIES OF ROTATION ABOUT A POINT
- If \latex{\alpha \neq k\times 360º(k\in \Z)} then the only fixed point of the transformation is its centre \latex{O}. If \latex{\alpha = k\times 360º(k\in \Z)} then every point in the plane is a fixed point (identical transformation).
- If \latex{\alpha = \pm 180º} then rotation about \latex{O} is the same as reflection in point \latex{O}. (Figure 41)
- Any rotation about a point is angle- and distance-preserving. (Figure 42)
\latex{-180\degree}
\latex{+180\degree}
\latex{ O }
\latex{ P }
\latex{ P' }
Figure 41
- The rotation about point \latex{O} by directed angle \latex{\alpha} can be obtained via two consecutive reflections across a line for which the intersection of the axes is \latex{O} and the angle between the axes is \latex{\frac{\mid\alpha\mid}{2}}. (Figure 43)
- Any rotation about a point is an orientation-preserving transformation.
\latex{ P }
\latex{ O }
\latex{ t_2 }
\latex{ t_1 }
\latex{ P'' }
\latex{ P' }
\latex{ P' }
\latex{ P }
\latex{ O }
\latex{ P'' }
\latex{ t_1 }
\latex{ t_2 }
Figure 43
\latex{AB= A’B’}
\latex{\alpha= \alpha’}
\latex{\alpha’}
\latex{\alpha}
\latex{ O }
\latex{ A }
\latex{ O }
\latex{ A' }
\latex{ B' }
\latex{ B }
Figure 42
ROTATIONALLY SYMMETRIC FIGURES
DEFINITION: A planar figure is rotationally symmetric, if there is point \latex{O} in the plane and there is an angle \latex{\alpha} with positive direction \latex{(0º \lt \alpha\lt 360º)}, so that this figure is an invariant figure when rotated about point \latex{O} by angle \latex{\alpha}.
IV. Parallel translation
DEFINITION: A vector \latex{\vec v} in the plane is given. To each point \latex{P} in the plane assign the point \latex{P’} for which \latex{\overrightarrow {PP’}= \vec v} .
\latex{\vec v} is called the vector of the parallel translation.
\latex{\vec v} is called the vector of the parallel translation.
A translation is uniquely determined by \latex{\vec v} or by a point \latex{P} and its image \latex{P’}.
PROPERTIES OF PARALLEL TRANSLATION
- Translations do not have fixed points if \latex{\vec v \neq \vec0} and every point is fixed if \latex{\vec v = \vec0} .
- Lines parallel with the direction of the translation are exactly the invariant lines.
- Any line is parallel to its image.
- Parallel translations are angle- and distance-preserving transformations.
- A parallel translation can be obtained by two consecutive reflections for which the axes are perpendicular to the direction of the translation and the distance of the axes is half of the length of the vector of the translation. (Figure 44)
- Parallel translations are orientation-preserving transformations.
\latex{ A }
\latex{ A' }
\latex{ A'' }
\latex{ C'' }
\latex{ C' }
\latex{ C }
\latex{ B }
\latex{ B' }
\latex{ B'' }
\latex{ y' }
\latex{ y' }
\latex{ y }
\latex{ y }
\latex{ z }
\latex{ z }
\latex{ z' }
\latex{ z' }
\latex{ x' }
\latex{ x' }
\latex{ x }
\latex{ x }
\latex{ t_1 }
\latex{ d }
\latex{ t_2 }
Figure 44
◆ ◆ ◆
*We did not study \latex{ 3 }-dimensional transformations during our studies, but obviously those can be examined similarly to the planar ones. Special \latex{ 3 }-dimensional transformations include
– reflection across a plane;
– reflection across an axis (across a line in space);
– reflection through a point;
– rotation about an axis;
– parallel translation.
– reflection across an axis (across a line in space);
– reflection through a point;
– rotation about an axis;
– parallel translation.
We will not define these, but it is worth noting that the definition of \latex{ 3 }-dimensional reflection through a point and translation coincide with the planar one.
Congruence of figures
DEFINITION: Two figures are congruent if there is a series of congruent transformations which transforms one of the figures into the other one. (Congruency is marked by \latex{\cong}.)
I. The basic cases of the congruence of triangles
THEOREM: Two triangles are congruent if and only if one of the following conditions applies:
- the length of the corresponding sides are mutually equal;
- the length of two sides of each triangle are mutually equal, and the angles included between these are equal;
- the length of one side in each triangle and the two angles on these sides are mutually equal;
- the length of two and two sides of the triangles are mutually equal, and the angles opposite the longer ones of these sides are equal.
SIMILARITY TRANSFORMATION
Parallel intercepting lines, parallel intercepting
line segments (higher level courseware)
line segments (higher level courseware)
*THEOREM: If we cut the arms of an angle with parallel straight lines, then the ratio of the intercepts resulting on one arm is equal to the ratio of the corresponding intercepts resulting on the other arm. (Intercept theorem.) (Figure 45)
\latex{ O }
\latex{ C' }
\latex{ D' }
\latex{ B }
\latex{ A }
\latex{ A' }
\latex{ B' }
\latex{ C }
\latex{ D }
\latex{ A' }
\latex{ B' }
\latex{ C' }
\latex{ D' }
\latex{ D }
\latex{ C }
\latex{ B }
\latex{ A }
\latex{ \frac{AB}{CD}=\frac{A'B'}{C'D’} }
\latex{ \frac{OA}{OB}=\frac{OA'}{OB’} }
Figure 45
*THEOREM: If two straight lines cut such line segments from the arms of an angle, the ratio of the length of which – measured from the vertex – is the same on both arms, then the two straight lines are parallel. (Reverse of the intercept theorem.)
\latex{ A }
\latex{ B }
\latex{ B' }
\latex{ b }
\latex{ a }
\latex{ A' }
\latex{ O }
Figure 46
Following the notation of Figure 46 these two theorems combined mean that
\latex{a \space{\text{and}} \space b \space\text{are parallel}\Leftrightarrow \frac{OA}{OB}=\frac{OA’}{OB’} .}
*THEOREM: If the arms of an angle are intersected by parallel lines, then the ratio of the segments of the lines determined by the arms of the angle equals the ratio of the segments on the arms of the angle, that is (Figure 47):
\latex{\frac{AA’}{BB’}=\frac{OA}{OB}=\frac{OA’}{OB’} .}
(Intercept theorem for line segments.)
\latex{ A }
\latex{ B }
\latex{ B' }
\latex{ A' }
\latex{ O }
\latex{ a }
\latex{ b }
Figure 47
Transformation of central dilation (or homothety)
DEFINITION: A point \latex{O} and a real number \latex{\lambda (\lambda \neq 0)} is given. Assign a point \latex{P’} to every point \latex{P} in the space as follows:
– if \latex{P = O}, then \latex{P’= P};
– if \latex{P\neq O}, then \latex{P’} is that point of the line \latex{OP} for which \latex{OP’ = |\lambda|\times OP} and if \latex{\lambda \gt 0} then \latex{P’} is a point on the ray \latex{OP} while if \latex{\lambda \lt 0} then \latex{P} and \latex{P’} are separated from each other by \latex{O}. (Figure 48)
– if \latex{P = O}, then \latex{P’= P};
– if \latex{P\neq O}, then \latex{P’} is that point of the line \latex{OP} for which \latex{OP’ = |\lambda|\times OP} and if \latex{\lambda \gt 0} then \latex{P’} is a point on the ray \latex{OP} while if \latex{\lambda \lt 0} then \latex{P} and \latex{P’} are separated from each other by \latex{O}. (Figure 48)
\latex{\lambda=\frac{1}{3}}
\latex{\lambda=-2}
\latex{ O }
\latex{ P' }
\latex{ P }
\latex{ P }
\latex{ O }
\latex{ P' }
Figure 48
The point \latex{O} is the centre of the transformation, the \latex{\lambda} is the scale factor or ratio of the central dilation.
If \latex{ \mid\lambda\mid \lt 1}, then we are talking about central reduction, if \latex{ \mid\lambda\mid \gt 1}, then about central enlargement.
The central dilation is unambiguously defined by its centre \latex{O} and the scale factor \latex{\lambda}, or by \latex{O} and a point different from \latex{O} together with its image.
PROPERTIES OF CENTRAL DILATION
- If \latex{\lambda\neq1}, then the sole fixed point of the dilation is the centre \latex{O}. If \latex{\lambda=1}, then every point of the space is fixed, that is, the transformation is the identical transformation.
- The straight lines passing through the point \latex{O} are the invariant straight lines of the transformation. If \latex{\lambda\neq1}, then there are no other invariant lines.
- The image of any straight line not containing the point \latex{O} is a straight line not passing through \latex{O} and parallel to the original line.
- Central dilation is angle-preserving transformation.
- For a scaling with ratio \latex{\lambda} the length of the image of an arbitrary segment equals the length of the original segment times \latex{\mid\lambda\mid}, that is, for arbitrary points \latex{A} and \latex{B}, \latex{A’B’=|\lambda|\times AB}.
- Scaling is an isometry if and only if \latex{\mid\lambda\mid=1}. If \latex{\lambda=1}, then it is the identical transformation, while if \latex{\lambda=-1}, then it is the point reflection.
- Central dilation in the plane is an orientation-preserving transformation.
Similarity transformation
DEFINITION: The transformations obtained as the result of a central dilation and some isometries are called similarity transformations.
PROPERTIES OF SIMILARITY TRANSFORMATIONS
- The image of any straight line under a similarity transformation is a straight line.
- Similarity transformations are angle-preserving transformations.
- For any similarity with ratio \latex{\lambda} and for any points \latex{A, B} and their images \latex{A’} and \latex{B’} it follows that \latex{\frac{A’B’}{AB}=|\lambda|}.
- If \latex{|\lambda|= 1}, than the similarity transformation is an isometry.
Similarity of figures
DEFINITION: The figures are said to be similar if there exists a similarity transformation which maps one object to the other. (Similarity is marked by \latex{\sim}.)
I. The simple cases of similar triangles
THEOREM: Two triangles are similar if and only if one of the following is satisfied:
- the ratio of the length of corresponding sides are equal;
- the ratio of the length of two pairs of sides are equal and the angle between these sides is equal;
- two pairs of angles are equal;
- the ratio of the length of two pairs of sides are equal and the angles not opposite the longer sides are equal.
II. Similarity of polygons
*THEOREM: Two polygons are similar if and only if the ratio of the length of corresponding sides are equal and their angles are equal.
III. Ratio of area for similar planar figures, ratio of volume for similar \latex{ 3 }-dimensional figures
DEFINITION: The ratio of two objects' similarity is the ratio of the lengths of the corresponding segments.
THEOREM: The ratio of the area of similar planar figures equals the square of the ratio of their similarity:
\latex{\frac{T’}{T}=\lambda^2}.
THEOREM: The ratio of the volume of similar 3-dimensional figures equals the cube of the ratio of their similarity:
\latex{\frac{V’}{V}=\lambda^3}.
Exercises
{{exercise_number}}. In the plane, two intersecting lines are given with a point not containing any of them. Construct a triangle for which the point is one of its vertices and the two lines are the bisectors of the two sides meeting in that vertex.
{{exercise_number}}. Draw a square in the coordinate system, which is symmetric for both axes and satisfies that the sum of the absolute values of its coordinates equals \latex{64}. How many solutions are there?
{{exercise_number}}. Construct a regular triangle from its orthocentre and one vertex.
{{exercise_number}}. Take a regular triangle with side length of \latex{4\,cm}, and extend it in the direction of one of its altitude lines by the \latex{\frac{2}{3}} part of the altitude. What planar figure is the
- intersection;
- union
of the original and the image of the triangle? Determine the circumference and area of the figure in both cases.
{{exercise_number}}. Prove that a perpendicular to the bisector of an angle determine segments of equal length on the arms of the angle.
{{exercise_number}}. Divide a given segment into \latex{7} equal parts.
{{exercise_number}}. The vertices of a triangle in the coordinate system are \latex{A(-2; 5), B(1; -3), C(8; 2)}. Dilate the triangle to twice as big using
- the origin as centre;
- the point \latex{O(6; -2)} as centre.
Determine the vertices of the image in both cases.
{{exercise_number}}. From a rectangular slide of size \latex{18\times25\,mm}, the projector creates an undistorted, sharp image with an area of \latex{2.88\,m^2} on the screen. Find the ratio of dilation. How far is the slide from the image on the screen if the distance between the slide and the bulb (which we can consider as a point) is \latex{5\,cm}, the bulb is on the line perpendicular to the slide which intersects it in the centre and the projection can be considered as a dilation with centre being the bulb?