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Construct an affine geometry over a set of 4×4 = 16 points?
Do not be afraid of affine geometry.
1. Straight lines are sets of points.
2. Any two points define a straight line.
3. There is always one parallel straight line, all other straight lines intersect at one point.
Bonus question:
why there are no circles here?
Lets do it for a p=3, a set of 3×3 = 9 points.
To construct a geometry is to construct sub-sets of points and declare such sets “straight lines”
Lets take set of the three elments of the field modulo p=3 T={0,1,2} and introduce cartesian coordinates (x,y). Each of the 9 points now has coordinates (x,y), for example the point in the center is (x=1,y=1).
Straightforwardly lets draw straight lines as
y = ax + b.
There will be
three blue parallel straight lines for a = 0, and
three red parallel straight lines for a = 1, and
three black parallel straight lines for a = 2.
We also need 3 “vertical” green parallelstraight lines in the form of
x = b.
There are altogether 12 straight lines, each line contains 3 points.
The propreties of the field T assure that there is a line for any two points, as well as one parallel line. Becuse given that T is a field the equation
ax+b = cx+d
has exactly one solution unless a=b
Construction of case 4×4 = 16 is a bit different – because modulo 4 is not a field.
“BY NO MEANS do I want the answer, but I would like some insight and advice on this one to get me off on the right foot”
OK, I’ll give you a hint, then: If you have two distinct lines L1 and L2 in an affine plane, there must be at least one point a on L1 but not L2, and at least one point b on L2 but not L1. There is a line ab through these two points. Through each point of L1 other than a, there is one and only one parallel to ab.
Take a close, careful look at those parallel lines, and you’ll find your proof.
I found the same… the straightforward way breaks axiom 2 in the case where (Δx,Δy) is of order 2. Making a 4-element field, although possible, does not help as its characteristic is 2 and we don’t want 2-point lines.
So far, I found only a trivial geometry where there is only one line consisting of all the points. But that does not exclude the existence of another solution.
Edit: I think I found a solution inspired by the figure “Forming a non-brick octad” from Dr. Octavian’s link [1]. It’s listed below: there are 20 straight lines of length 4 arranged so that all lines in one row are parallel.
{{11,12,13,14}, {21,22,23,24}, {31,32,33,34}, {41,42,43,44},
{11,21,31,41}, {12,22,32,42}, {13,23,33,43}, {14,24,34,44},
{11,24,33,42}, {14,21,32,43}, {44,13,22,31}, {41,12,23,34},
{11,23,32,44}, {14,22,33,41}, {12,24,31,43}, {13,41,34,42},
{11,22,34,43}, {12,21,33,44}, {13,24,32,41}, {14,23,31,42}}
The first group are horizontal lines, then vertical lines. The third group was taken from the reference. There was only one possibility left how to complete this set.
Thank you, Dr. Octavian!
Edit2: It seems that the idea of the 4-element Galois field F4 [2] was not that bad… the straight lines can also be defined as y = ax+b in this field, plus the vertical lines x = const. I don’t know why did I think that this led to something wrong.
The elements of this field can be denoted 00, 01, 10 and 11. The addition is bit-wise XOR, 00 is the zero element and 01 the unit element. The rules of multiplication are:
00 * x = 00 for all x
01 * x = x for all x
10 * 10 = 11
10 * 11 = 01
11 * 11 = 10
& the multiplication is commutative.
Identifying 00, 01, 10 and 11 with 1, 2, 3 and 4, respectively, this gives the following set of straight lines:
{{11,12,13,14}, {21,22,23,24}, {31,32,33,34}, {41,42,43,44},
{11,21,31,41}, {12,22,32,42}, {13,23,33,43}, {14,24,34,44},
{11,22,33,44}, {12,21,34,43}, {13,24,31,42}, {14,23,32,41},
{11,23,34,42}, {12,24,33,41}, {13,21,32,44}, {14,22,31,43},
{11,24,32,43}, {12,23,31,44}, {13,22,34,41}, {14,21,33,42}}
geometry multiple choice?
1. In projective geometry, a projection is defined as a transformation of __________ from one plane to another. This is why projective geometry used to be called the geometry of position.
A.
measurements
B.
points and lines
C.
circles
D.
squares
2. Reflections, translations, and rotations are rigid transformations. They are also __________ transformations-transformations that preserve straight lines, ratios of distances, intersecting lines, and parallel lines.
A.
affine
B.
infinite
C.
finite
D.
approximate
3. Which theorem states that the intersection of the lines through the corresponding sides of a triangle and its projected image must be collinear?
A.
Pappus’s Theorem
B.
Desargues’ Theorem
C.
Pascal’s Theorem
D.
Pythagorean Theorem
please answer only if you think your positive and thankyou to all who answer!
College Geometry Question (With Respect to Affine Planes)?
“Show that any two lines in an affine plane have the same number of points (ie there exists a 1 to 1 correspondence between the points of the two lines.”
For those who reply to these types of questions by saying that I need to do my own homework…..BY NO MEANS do I want the answer, but I would like some insight and advice on this one to get me off on the right foot. Thanks!
Geometry Help Please!!! Multiple Choice?
Which mathematician is NOT known as a contributor to projective geometry?
Desargues
Pythagoras
Pappus
In projective geometry, a projection is defined as a transformation of __________ from one plane to another. This is why projective geometry used to be called the geometry of position.
measurements
points and lines
circles
squares
Reflections, translations, and rotations are rigid transformations. They are also __________ transformations—transformations that preserve straight lines, ratios of distances, intersecting lines, and parallel lines.
affine
infinite
finite
approximate
Which theorem states that the intersection of the lines through the corresponding sides of a triangle and its projected image must be collinear?
Pappus’s Theorem
Desargues’ Theorem
Pascal’s Theorem
Pythagorean Theorem
If you begin with a right triangle with vertices at A(1, 1), B(4, 1), and C(1, 5), and then transform the figure by using the transformation (x, y) = (2x, 3y), what are the coordinates of the transformed point B?
(2, 3)
To the last one the answer is (8, 3). To the one before that, the answer is NOT Pythagorean Theorem, I think the one before that is infinite and I have no clue to the other one.
Q2
The transformations that move lines into lines, while preserving their intersection properties, are special and interesting, because they will move all polylines into polylines and all polygons into polygons. These are the affine transformations.
Construct an affine geometry over a set of 4×4 = 16 points?
Do not be afraid of affine geometry.
1. Straight lines are sets of points.
2. Any two points define a straight line.
3. There is always one parallel straight line, all other straight lines intersect at one point.
Bonus question:
why there are no circles here?
Lets do it for a p=3, a set of 3×3 = 9 points.
http://img16.imageshack.us/img16/9315/3×3.gif
To construct a geometry is to construct sub-sets of points and declare such sets “straight lines”
Lets take set of the three elments of the field modulo p=3 T={0,1,2} and introduce cartesian coordinates (x,y). Each of the 9 points now has coordinates (x,y), for example the point in the center is (x=1,y=1).
Straightforwardly lets draw straight lines as
y = ax + b.
There will be
three blue parallel straight lines for a = 0, and
three red parallel straight lines for a = 1, and
three black parallel straight lines for a = 2.
We also need 3 “vertical” green parallelstraight lines in the form of
x = b.
There are altogether 12 straight lines, each line contains 3 points.
The propreties of the field T assure that there is a line for any two points, as well as one parallel line. Becuse given that T is a field the equation
ax+b = cx+d
has exactly one solution unless a=b
Construction of case 4×4 = 16 is a bit different – because modulo 4 is not a field.
How to obtain the matrix for a rotation about an arbitrary point?
I found the actual matrix here: http://www.euclideanspace.com/maths/geometry/affine/aroundPoint/index.htm
It’s a rotation through an angle theta about a point (a,b)
I don’t understand how he/she got this matrix? Help anyone!!!!
“BY NO MEANS do I want the answer, but I would like some insight and advice on this one to get me off on the right foot”
OK, I’ll give you a hint, then: If you have two distinct lines L1 and L2 in an affine plane, there must be at least one point a on L1 but not L2, and at least one point b on L2 but not L1. There is a line ab through these two points. Through each point of L1 other than a, there is one and only one parallel to ab.
Take a close, careful look at those parallel lines, and you’ll find your proof.
I found the same… the straightforward way breaks axiom 2 in the case where (Δx,Δy) is of order 2. Making a 4-element field, although possible, does not help as its characteristic is 2 and we don’t want 2-point lines.
So far, I found only a trivial geometry where there is only one line consisting of all the points. But that does not exclude the existence of another solution.
Edit: I think I found a solution inspired by the figure “Forming a non-brick octad” from Dr. Octavian’s link [1]. It’s listed below: there are 20 straight lines of length 4 arranged so that all lines in one row are parallel.
{{11,12,13,14}, {21,22,23,24}, {31,32,33,34}, {41,42,43,44},
{11,21,31,41}, {12,22,32,42}, {13,23,33,43}, {14,24,34,44},
{11,24,33,42}, {14,21,32,43}, {44,13,22,31}, {41,12,23,34},
{11,23,32,44}, {14,22,33,41}, {12,24,31,43}, {13,41,34,42},
{11,22,34,43}, {12,21,33,44}, {13,24,32,41}, {14,23,31,42}}
The first group are horizontal lines, then vertical lines. The third group was taken from the reference. There was only one possibility left how to complete this set.
Thank you, Dr. Octavian!
Edit2: It seems that the idea of the 4-element Galois field F4 [2] was not that bad… the straight lines can also be defined as y = ax+b in this field, plus the vertical lines x = const. I don’t know why did I think that this led to something wrong.
The elements of this field can be denoted 00, 01, 10 and 11. The addition is bit-wise XOR, 00 is the zero element and 01 the unit element. The rules of multiplication are:
00 * x = 00 for all x
01 * x = x for all x
10 * 10 = 11
10 * 11 = 01
11 * 11 = 10
& the multiplication is commutative.
Identifying 00, 01, 10 and 11 with 1, 2, 3 and 4, respectively, this gives the following set of straight lines:
{{11,12,13,14}, {21,22,23,24}, {31,32,33,34}, {41,42,43,44},
{11,21,31,41}, {12,22,32,42}, {13,23,33,43}, {14,24,34,44},
{11,22,33,44}, {12,21,34,43}, {13,24,31,42}, {14,23,32,41},
{11,23,34,42}, {12,24,33,41}, {13,21,32,44}, {14,22,31,43},
{11,24,32,43}, {12,23,31,44}, {13,22,34,41}, {14,21,33,42}}
What is a circle defined like, please?
geometry multiple choice?
1. In projective geometry, a projection is defined as a transformation of __________ from one plane to another. This is why projective geometry used to be called the geometry of position.
A.
measurements
B.
points and lines
C.
circles
D.
squares
2. Reflections, translations, and rotations are rigid transformations. They are also __________ transformations-transformations that preserve straight lines, ratios of distances, intersecting lines, and parallel lines.
A.
affine
B.
infinite
C.
finite
D.
approximate
3. Which theorem states that the intersection of the lines through the corresponding sides of a triangle and its projected image must be collinear?
A.
Pappus’s Theorem
B.
Desargues’ Theorem
C.
Pascal’s Theorem
D.
Pythagorean Theorem
please answer only if you think your positive and thankyou to all who answer!
College Geometry Question (With Respect to Affine Planes)?
“Show that any two lines in an affine plane have the same number of points (ie there exists a 1 to 1 correspondence between the points of the two lines.”
For those who reply to these types of questions by saying that I need to do my own homework…..BY NO MEANS do I want the answer, but I would like some insight and advice on this one to get me off on the right foot. Thanks!
Geometry Help Please!!! Multiple Choice?
Which mathematician is NOT known as a contributor to projective geometry?
Desargues
Pythagoras
Pappus
In projective geometry, a projection is defined as a transformation of __________ from one plane to another. This is why projective geometry used to be called the geometry of position.
measurements
points and lines
circles
squares
Reflections, translations, and rotations are rigid transformations. They are also __________ transformations—transformations that preserve straight lines, ratios of distances, intersecting lines, and parallel lines.
affine
infinite
finite
approximate
Which theorem states that the intersection of the lines through the corresponding sides of a triangle and its projected image must be collinear?
Pappus’s Theorem
Desargues’ Theorem
Pascal’s Theorem
Pythagorean Theorem
If you begin with a right triangle with vertices at A(1, 1), B(4, 1), and C(1, 5), and then transform the figure by using the transformation (x, y) = (2x, 3y), what are the coordinates of the transformed point B?
(2, 3)
(2, 1/3)
(8, 3)
(12, 2)
To the last one the answer is (8, 3). To the one before that, the answer is NOT Pythagorean Theorem, I think the one before that is infinite and I have no clue to the other one.
Hope this helps!!
Q1
B.
points and lines
Q2
The transformations that move lines into lines, while preserving their intersection properties, are special and interesting, because they will move all polylines into polylines and all polygons into polygons. These are the affine transformations.
Q3
not D ore C