CHAPTER 1. LINES AND PLANES IN SPACE §1. Angles and

... 1.13.Through point O2 , draw line l10 parallel to l1 . Let Π be the plane containing lines l2 and l10 ; A01 the projection of point A1 to plane Π. As follows from Problem 1.11, line A01 A2 constitutes equal angles with lines l10 and l2 and, therefore, triangle A01 O2 A2 is an equilateral one, hence, ...

... 1.13.Through point O2 , draw line l10 parallel to l1 . Let Π be the plane containing lines l2 and l10 ; A01 the projection of point A1 to plane Π. As follows from Problem 1.11, line A01 A2 constitutes equal angles with lines l10 and l2 and, therefore, triangle A01 O2 A2 is an equilateral one, hence, ...

CHAPTER 7 Pictorial Projections

... Oblique projection is a unique form of parallel projection. As the name indicates, oblique projection results when the projectors are parallel to each other but at some angle other than perpendicular to the projection plane. If the principal view of the object is placed such that its surfaces are pa ...

... Oblique projection is a unique form of parallel projection. As the name indicates, oblique projection results when the projectors are parallel to each other but at some angle other than perpendicular to the projection plane. If the principal view of the object is placed such that its surfaces are pa ...

Spherical Geometry Activities - Notes

... the two boundary points, because they count the boundaries in the partitioning.] One of the sections is finite and the other two are infinite. 4) How do you measure the distance between two points on the plane? Measure the distance between A and B. Is there another way to measure the distance betwee ...

... the two boundary points, because they count the boundaries in the partitioning.] One of the sections is finite and the other two are infinite. 4) How do you measure the distance between two points on the plane? Measure the distance between A and B. Is there another way to measure the distance betwee ...

“Straight” and “Angle” on Non-Planar Surfaces

... At the teacher’s option, each group could build a paper soccer ball using hexagons and pentagons. This can be done using the template on the next page and leaving the pentagonal holes empty. T: Now, what would happen if we started with a seven-sided polygon (called a heptagon) and surrounded it with ...

... At the teacher’s option, each group could build a paper soccer ball using hexagons and pentagons. This can be done using the template on the next page and leaving the pentagonal holes empty. T: Now, what would happen if we started with a seven-sided polygon (called a heptagon) and surrounded it with ...

axonometric plane - Design Communication Graphics

... What is Axonometric Projection? • Axonometric Projection is a parallel projection technique used to create a pictorial drawing of an object by projecting that object onto a plane • The plane of projection is called the axonometric plane • When the projectors are drawn perpendicular to the axonometr ...

... What is Axonometric Projection? • Axonometric Projection is a parallel projection technique used to create a pictorial drawing of an object by projecting that object onto a plane • The plane of projection is called the axonometric plane • When the projectors are drawn perpendicular to the axonometr ...

An Elementary Introduction to the Hopf Fibration

... squares of the three coordinates on the right hand side sum to (a 2 +b2 +c2 +d2 )2 = 1, so that the image of h is indeed contained in S 2 . What problem was Hopf trying to solve when he invented this map? And how can one see any connection with physical rotations, as we have claimed? The work in Hop ...

... squares of the three coordinates on the right hand side sum to (a 2 +b2 +c2 +d2 )2 = 1, so that the image of h is indeed contained in S 2 . What problem was Hopf trying to solve when he invented this map? And how can one see any connection with physical rotations, as we have claimed? The work in Hop ...

Name - TeacherWeb

... *Use your paper as a plane and your pencil as a line* A set of three non-collinear points determine a plane. A plane containing any two points contains all of the points on the line determine by those two points. There is exactly one plane containing a line and a point not on the line. If tw ...

... *Use your paper as a plane and your pencil as a line* A set of three non-collinear points determine a plane. A plane containing any two points contains all of the points on the line determine by those two points. There is exactly one plane containing a line and a point not on the line. If tw ...

A Horopter for Two - Smith-Kettlewell Eye Research Institute

... Perspective is an exact construction of the scene viewed from a particular location ...

... Perspective is an exact construction of the scene viewed from a particular location ...

Homothetic centers of three circles and their three

... places from where three vertices look like collinear, however, there are at most two places from where three angles of elevation are equal to each other. In section 2, we review an easy construction of homothetic centers of two circles. Using the homothetic centers, we will construct the points fro ...

... places from where three vertices look like collinear, however, there are at most two places from where three angles of elevation are equal to each other. In section 2, we review an easy construction of homothetic centers of two circles. Using the homothetic centers, we will construct the points fro ...

Map Projections And State Plane Coordinate Systems

... Map Projections and State Plane Coordinate Systems A Sun Cam online continuing education course 4) Shape/Conformity: This projection concerns itself with the preservation of the shape. It is used for topographic mapping and navigation. Its advantages are that every parallel line intersects with eve ...

... Map Projections and State Plane Coordinate Systems A Sun Cam online continuing education course 4) Shape/Conformity: This projection concerns itself with the preservation of the shape. It is used for topographic mapping and navigation. Its advantages are that every parallel line intersects with eve ...

Perspective Projection

... • Note that all points on the line defined by (xmin,ymin,zmin) and COP are mapped to the lower left point on the viewport. • Also all points on the line defined by (xmax,ymax,-zmin) and COP are mapped to the upper right corner of the viewport. • The viewing direction is always parallel to -z • It is ...

... • Note that all points on the line defined by (xmin,ymin,zmin) and COP are mapped to the lower left point on the viewport. • Also all points on the line defined by (xmax,ymax,-zmin) and COP are mapped to the upper right corner of the viewport. • The viewing direction is always parallel to -z • It is ...

Projective Geometry

... If L and M are any two lines, then they meet at a single point. Case 1: L and M are ordinary, non-parallel lines: as usual. Case 2: L and M are ordinary, parallel lines: they meet at the corresponding point at infinity. Case 3: L is an ordinary line and M is the line at infinity: they meet at the po ...

... If L and M are any two lines, then they meet at a single point. Case 1: L and M are ordinary, non-parallel lines: as usual. Case 2: L and M are ordinary, parallel lines: they meet at the corresponding point at infinity. Case 3: L is an ordinary line and M is the line at infinity: they meet at the po ...

Overview - Connecticut Core Standards

... Students generally think of a sphere as a solid (for example a baseball or a basketball). In this investigation we want them to focus on the surface. One way to get them to see this is to view the sphere as a solid of rotation. This idea was first introduced in question 10 of Activity 6.4.2 where st ...

... Students generally think of a sphere as a solid (for example a baseball or a basketball). In this investigation we want them to focus on the surface. One way to get them to see this is to view the sphere as a solid of rotation. This idea was first introduced in question 10 of Activity 6.4.2 where st ...

Geography English online Hotips.pmd - LD Clerk

... • The wind that causes sun-stroke - Loo • A strong wind which develops on the leeward side of the Alps - Foehn • A wind that hastens the ripening of grapes and helps to melt snow rapidly in the Northern side of ...

... • The wind that causes sun-stroke - Loo • A strong wind which develops on the leeward side of the Alps - Foehn • A wind that hastens the ripening of grapes and helps to melt snow rapidly in the Northern side of ...

Math 420 - Homework 3

... (b) There is a unique line passing through any two distinct points. (c) A straight line never ends. (d) If three points are collinear, one point is between the other two. (e) Perpendicular lines form four right angles. 3. Describe the three ways that a plane P and a sphere S with center C may inters ...

... (b) There is a unique line passing through any two distinct points. (c) A straight line never ends. (d) If three points are collinear, one point is between the other two. (e) Perpendicular lines form four right angles. 3. Describe the three ways that a plane P and a sphere S with center C may inters ...

13 A Glimpse at Elliptic Geometry

... D = Ba on the equator ∂D. We connect them by two diﬀerent half great circles a and c, lying symmetric to the south pole O. Next draw a segment through O that cuts c at point A and a at point C, but not at a right angle. Nevertheless, you get a ﬁgure which is point symmetric by point O. Hence the tri ...

... D = Ba on the equator ∂D. We connect them by two diﬀerent half great circles a and c, lying symmetric to the south pole O. Next draw a segment through O that cuts c at point A and a at point C, but not at a right angle. Nevertheless, you get a ﬁgure which is point symmetric by point O. Hence the tri ...

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... (1) Section 5.1: Inversion. We developed the concept of inversion in (or through) a circle. We found that this operation is self-inverse, and that it generalizes reflection over a line. We also found that what we now call generalized circles (circles or lines) get mapped to generalized circles (depe ...

... (1) Section 5.1: Inversion. We developed the concept of inversion in (or through) a circle. We found that this operation is self-inverse, and that it generalizes reflection over a line. We also found that what we now call generalized circles (circles or lines) get mapped to generalized circles (depe ...

TWO CAMERAS 2009

... The investigation of image formation and object modeling is widely developed in the last thirty years. The formation of planar images of our three dimensional world plays an important role in our century of communications through computers. The geometry of multiple images provides us the description ...

... The investigation of image formation and object modeling is widely developed in the last thirty years. The formation of planar images of our three dimensional world plays an important role in our century of communications through computers. The geometry of multiple images provides us the description ...

Stereonets

... Flatten the sphere to two dimensions by projecting the lower hemisphere intersections to an equatorial plane of reference that passes through the center of the sphere. Lower hemisphere intersections are projected as rays upwards through the horizontal reference plane to the zenith of the sphere ...

... Flatten the sphere to two dimensions by projecting the lower hemisphere intersections to an equatorial plane of reference that passes through the center of the sphere. Lower hemisphere intersections are projected as rays upwards through the horizontal reference plane to the zenith of the sphere ...

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet or Wulff net.