Prove that an affine plane extended by ideal points and ideal line will satisfy projective Axiom ...
prove that an affine plane extended by ideal points and ideal line will satisfy projective Axiom 3. (Use cases: 1) when two lines are affine lines with an ideal point added 2) when of one the two lines is an ideal line)
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Prove that an affine plane extended by ideal points and ideal line will satisfy projective Axiom ...
using these axioms prove proof number 5 1 - . Axiom 1: There exist at least...
using these axioms prove proof number 5
1 - . Axiom 1: There exist at least one point and at least one line Axiom 2: Given any two distinct points, there is exactly one line incident with both points Axiom 3: Not all points are on the same line. Axiom 4: Given a line and a point not on/ there exists exactly one linem containing Pouch that / is parallel tom Theorem 1: If two distinct lines are not parallet,...
.3. Let A and B be distinct points. Prove that for each real number r E (-00, oo) there is exactl...
.3. Let A and B be distinct points. Prove that for each real number r E (-00, oo) there is exactly one point on the extended line AB such that AX/XB- r. Which point on AB does not correspond to any real number r? 4. Draw an example of a triangle in the extended Euclidean plane that has one ideal vertex. Is there a triangle in the extended plane that has two ideal vertices? Could there be a triangle with...
With the given notes from below, answer number one please Axiom 1: there exists at least one line...
With the given notes from below, answer number one please Axiom 1: there exists at least one line. Axiom 2: every line has exactly 8 points incident (passes through) to it Axiom 3: not all points are incident to the same line Axiom 4: there is no line containing all points Axiom 5: there is at least two points on one line Axiom 6: there exists at least two lines Axiom 7: there is exactly on with incident with any...
Duality Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct...
Duality
Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct lines. Axiom 3. There is exactly 1 line with exactly 3 distinct points on it. Axiom 4. Given any 2 distinct points, there exists at least 1 line passing through the 2 points. Which of the following is the dual of Axiom 4? O a. Every line has at least 2 points on it. b. There exists at least 1 point with at least...
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are...
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are parallel if they have no point in common. Axioms: I. Given any two distinct points, there is a unique line on both. II. Each line has at least two distinct points on it. III. There exist at least three points. IV. Not all points...
3 Remember that for projective geometry, the dual of a statement is found by exchanging "point...
3 Remember that for projective geometry, the dual of a statement is found by exchanging "point with "ie" (a) Write the dual of the following statement, and then sketch picturess- trating both the statement and its dual. "Two distinct points are on one and only one line." (b) Draw a triangle, then draw its dual. (c) Draw a picture (collection of points and lines) which is not its own dual.
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove...
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove that if the converse of the Alternate Interior Angle Theorem is true, then all triangles have zero defect. [Hint: For an arbitrary triangle, ABC, draw a line through C parallel to side AB. Justify why you can do this.] 5) Consider the following statements: I: If two triangles are congruent, then they have equal defect. II: If two triangles are similar, then they have...
(a) Consider two lines on a plane in the homogeneous coordinate system l1 = (8,3,4)T and...
(a) Consider two lines on a plane in the homogeneous coordinate system l1 = (8,3,4)T and 12 = (5, -7,1)T. Where do they intersect? (b) Where is the ideal point corresponding to li? (c) Where would the ideal point from part (b) be located under the following projective transformation given by matrix H? 1 3 H= | -4 | 6 7 5 8 2 1 9
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional a...
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with...
4. (a) In a projective plane of order n, a set of k points with no three on the same line, is called a k-arc. Show that a k-arc has size at most n +2 [10 marks (b) An (n +2)-arc is called a hyper...
4. (a) In a projective plane of order n, a set of k points with no three on the same line, is called a k-arc. Show that a k-arc has size at most n +2 [10 marks (b) An (n +2)-arc is called a hyperoval. Show that a necessary condition for the existence of hyperovals is that n is even. 15 marks)
4. (a) In a projective plane of order n, a set of k points with no three on...
using these axioms prove proof number 5
1 - . Axiom 1: There exist at least one point and at least one line Axiom 2: Given any two distinct points, there is exactly one line incident with both points Axiom 3: Not all points are on the same line. Axiom 4: Given a line and a point not on/ there exists exactly one linem containing Pouch that / is parallel tom Theorem 1: If two distinct lines are not parallet,...
.3. Let A and B be distinct points. Prove that for each real number r E (-00, oo) there is exactly one point on the extended line AB such that AX/XB- r. Which point on AB does not correspond to any real number r? 4. Draw an example of a triangle in the extended Euclidean plane that has one ideal vertex. Is there a triangle in the extended plane that has two ideal vertices? Could there be a triangle with...
Duality
Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct lines. Axiom 3. There is exactly 1 line with exactly 3 distinct points on it. Axiom 4. Given any 2 distinct points, there exists at least 1 line passing through the 2 points. Which of the following is the dual of Axiom 4? O a. Every line has at least 2 points on it. b. There exists at least 1 point with at least...
Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are parallel if they have no point in common. Axioms: I. Given any two distinct points, there is a unique line on both. II. Each line has at least two distinct points on it. III. There exist at least three points. IV. Not all points...
3 Remember that for projective geometry, the dual of a statement is found by exchanging "point with "ie" (a) Write the dual of the following statement, and then sketch picturess- trating both the statement and its dual. "Two distinct points are on one and only one line." (b) Draw a triangle, then draw its dual. (c) Draw a picture (collection of points and lines) which is not its own dual.
(a) Consider two lines on a plane in the homogeneous coordinate system l1 = (8,3,4)T and 12 = (5, -7,1)T. Where do they intersect? (b) Where is the ideal point corresponding to li? (c) Where would the ideal point from part (b) be located under the following projective transformation given by matrix H? 1 3 H= | -4 | 6 7 5 8 2 1 9
3) Let us build a geometry, S, using the three axioms of incidence geometry with one additional axiom added: Incidence Axiom l: For every point P and every point Q (P and Q not equal), there exists a unique line, I, incident with P and Q. Incidence Axiom 2: For every line / there exist at least two distinct points incident with Incidence Axiom3: There exist (at least) three distinct points with the property that no line is incident with...
4. (a) In a projective plane of order n, a set of k points with no three on the same line, is called a k-arc. Show that a k-arc has size at most n +2 [10 marks (b) An (n +2)-arc is called a hyperoval. Show that a necessary condition for the existence of hyperovals is that n is even. 15 marks)
4. (a) In a projective plane of order n, a set of k points with no three on...
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