9-1
... if two points lie in a plane, then the line containing the points lies in the plane if two distinct planes intersect, then their intersection is a line there is exactly one plane that contains any three distinct noncollinear (not all on the same line) points a line and a point not on the lin ...
... if two points lie in a plane, then the line containing the points lies in the plane if two distinct planes intersect, then their intersection is a line there is exactly one plane that contains any three distinct noncollinear (not all on the same line) points a line and a point not on the lin ...
Unit Title: Suggested Time
... G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically (i.e. ,prove or disprove that the point (1, 3 ...
... G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically (i.e. ,prove or disprove that the point (1, 3 ...
Do Now
... • We would have to use the terms point, line, and plane to define them. • So we accept them as undefined and use them to define everything else in Geometry. ...
... • We would have to use the terms point, line, and plane to define them. • So we accept them as undefined and use them to define everything else in Geometry. ...
1-1
... Points, lines, and planes are called ________________terms. This means that they can only be explained using examples and descriptions. In the picture to the right, circle a point in RED. What is the name of the point? Circle a line in BLUE. What is the name of the line? Circle a plane in GREEN. W ...
... Points, lines, and planes are called ________________terms. This means that they can only be explained using examples and descriptions. In the picture to the right, circle a point in RED. What is the name of the point? Circle a line in BLUE. What is the name of the line? Circle a plane in GREEN. W ...
Day 1 Points Lines and Planes Continued
... An ___________________ is formed when two rays unite at their endpoints. The size of the angle is determined by the _______________________________________ between the two rays. ...
... An ___________________ is formed when two rays unite at their endpoints. The size of the angle is determined by the _______________________________________ between the two rays. ...
The PDF of our notes about Kant and Euclidean Geometry
... From The Cri+que of Pure Reason Take, for instance, the proposiDon, “Two straight lines cannot enclose a space, and with them alone no figure is possible”, and try to derive it from the concept o ...
... From The Cri+que of Pure Reason Take, for instance, the proposiDon, “Two straight lines cannot enclose a space, and with them alone no figure is possible”, and try to derive it from the concept o ...
Unit 1
... At the end of Unit student’s should be able to say “I can…” • use and understand definitions of angles, circles, perpendicular lines, parallel lines, and line segments based on the undefined terms of point, line, distance along a line and length of an arc. • describe and compare function transformat ...
... At the end of Unit student’s should be able to say “I can…” • use and understand definitions of angles, circles, perpendicular lines, parallel lines, and line segments based on the undefined terms of point, line, distance along a line and length of an arc. • describe and compare function transformat ...
4. G.1 Draw points, lines, line segments, rays, angles (right, acute
... Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. About the Math Points, lines, segments, rays, and angles are the building blocks of the geometry. Point and line are undefined terms because they d ...
... Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. About the Math Points, lines, segments, rays, and angles are the building blocks of the geometry. Point and line are undefined terms because they d ...
Lie sphere geometry
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is that lines (or planes) should be regarded as circles (or spheres) of infinite radius and that points in the plane (or space) should be regarded as circles (or spheres) of zero radius.The space of circles in the plane (or spheres in space), including points and lines (or planes) turns out to be a manifold known as the Lie quadric (a quadric hypersurface in projective space). Lie sphere geometry is the geometry of the Lie quadric and the Lie transformations which preserve it. This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres).To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.Lie sphere geometry can be defined in any dimension, but the case of the plane and 3-dimensional space are the most important. In the latter case, Lie noticed a remarkable similarity between the Lie quadric of spheres in 3-dimensions, and the space of lines in 3-dimensional projective space, which is also a quadric hypersurface in a 5-dimensional projective space, called the Plücker or Klein quadric. This similarity led Lie to his famous ""line-sphere correspondence"" between the space of lines and the space of spheres in 3-dimensional space.