2 edition of **Notes on stereographic projection and the astronomical triangle** found in the catalog.

Notes on stereographic projection and the astronomical triangle

Hendrickson, William Woodbury, 1844-1915.

- 381 Want to read
- 20 Currently reading

Published
**1905** by U.S. Naval institute in Annapolis .

Written in English

- Spherical astronomy.,
- Spherical projection.

**Edition Notes**

Statement | prepared by Professor W. W. Hendrickson |

Classifications | |
---|---|

LC Classifications | QB147 .H49 |

The Physical Object | |

Pagination | 31 p. |

Number of Pages | 31 |

ID Numbers | |

Open Library | OL6954811M |

LC Control Number | 05013172 |

OCLC/WorldCa | 17788500 |

4. Stereographic Projection There are two special projections: one onto the x-axis, the other onto the y-axis. Both are well-known. Using those projections one can deﬂne functions sine and cosine. However, there is another projection, less known to students, a projection from a circle to the x-axis. It is called the stereographic Linear projection maps altitude lines linearly along the radius of a sky-dome map or the y-axis of an orthographic map. Stereographic projection is a little more complex, but you can read more details about it on Wikipedia. Display Options. You can use the OPTIONS button menu (F4) to change the main display settings. This menu lets you toggle. a map called the Stereographic Projection. Instructions: (a) (10 points) Write a page paper on the stereographic projection. The paper should be typed, font-s and single-spaced. Be sure to include: Some historical background and motivation The mathematical signi cance of the projection A description of any real-world applications. Azimuth: A treatise on this subject, with a study of the astronomical triangle and of the effect of errors in the data. Illustrated by loci of maximum and minimum errors [Craig, Joseph Edgar] on *FREE* shipping on qualifying offers. Azimuth: A treatise on this subject, with a study of the astronomical triangle and of the effect of errors in the data.

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Notes on stereographic projection and the astronomical triangle, [William Woodbury Hendrickson] on *FREE* shipping on qualifying : William Woodbury Hendrickson. Title: Notes on stereographic projection and the astronomical triangle: Authors: HENDRICKSON, WILLIAM WOODBURY: Publication: Annapolis, U.

Naval institute, Notes on stereographic projection and the astronomical triangle, prepared by Professor W. Hendrickson c The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the was originally known as the planisphere projection.

Planisphaerium by Ptolemy is the oldest surviving document that describes it. One of its most important uses was the representation of celestial charts.

The term planisphere is still used to refer to such charts. • Stereographic projection can be used to measure the angle between any two lines.

• First the lines are plotted and then then the overlay is rotated until these two points lie on the same great circle of the stereonet and the angle between the two lines is determined by counting the small circle divisions between the points along the great.

Stereographic projection (p.m. Aug ) 6 C We want to show that the section of the cone by this plane is a circle. C A′ B′ a b P If P is any point of this intersection and P′ is the foot of the perpendicular from P to A′B′, we must show that PP′2 = A′P′ P′B′.Pass a plane parallel to the original one through the line PP′, and let a and b be the points on.

One approach might be using the fact that Möbius transformations are conformal (if you know that), and you can show that a stereographic projection onto the sphere combined with a rotation of the sphere combined with a stereographic projection back to the plane will result in a.

The term is loosely used to refer to any clock that shows, in addition to the time of day, astronomical information. This could include the location of the sun and moon in the sky, the age and Lunar phases, the position of the sun on the ecliptic and the current zodiac sign, the sidereal time, and other astronomical data such as the moon's nodes (for indicating eclipses) or a.

The Stereographic Projection E. Whittaker 1. The Purpose of the Stereographic Projection in Crystallography The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The projection is defined as shown in Fig.

An easy way to get intuition for this is to note that those formulas for the stereographic projection give equations for the point on the unit sphere (which you've labeled as $(x_1, x_2, x_3)$) if you draw a line through the north pole of the sphere (i.e.

$(x_1, x_2, x_3) =. Notes on stereographic projection and the astronomical triangle, (Annapolis, U.S. Naval institute, ), by William Woodbury Hendrickson (page images at HathiTrust) An introduction to practical astronomy: with a collection of astronomical tables. South Poles as defined in the projection above.

However, when plotting directional data in structural geology, they do represent the North and South geographic directions. As defined in our projection, the N and S poles would plot directly above and below the center of the stereonet. Stereographic Projection of Crystal Faces Page 3 of 6 9/7/ Proof that stereographic projection preserves circles.

As mentioned above, stereographic projection has two important characteristics. One being that stereographic projection preserves angles and the other being that stereographic projection preserves circles.

We now include a proof of this fact done in illustrations as well as an algebraic proof. Notes on stereographic projection and the astronomical triangle, (Annapolis, U.S. Naval institute, ), by William Woodbury Hendrickson (page images at HathiTrust) An introduction to practical astronomy: with a collection of astronomical tables.

(New York: Harper & Brothers, ), by Elias Loomis (page images at HathiTrust). 2 STEREOGRAPHIC PROJECTION IS CONFORMAL Stereographic projection is conformal, meaning that it preserves angles between curves. To see this, take a point p ∈ S2 \ {n}, let Tp denote the tangent plane to S2 at p, and let Tn denote File Size: 58KB.

stereographic projection has the property that all circles on the sphere are mapped onto circles or straight lines on the plane, and therefore it is easy to map astronomical observations.

We include a construction in Section3. Up to the late 18th century the Mercator and stereographic projections were treated as completely unrelated. The stereographic projection is conformal, i.e., the image of an angle on the sphere is an angle in $ of the same size.

and II. establish the the stereographic projection has the desired mapping properties. The proofs are based on the following Lemma. The image of a tangent segment to a point P on the sphere between P and $ isFile Size: 82KB.

I.4 Stereographic Projection (Not Examinable!) A complex number z = x+iy ∈C can be represented as point (x,y) in the plane R2. One can also associate a point (u,v,w) on the unit sphere S = {(u,v,w) ∈R3 |u2 +v2 +w2 = 1}, called the, with a given point (x,y) in the plane.

The associated mapping is called stereographic projection. • •. Lecture Notes - Mineralogy - Stereographic Projections • The stereographic projection is a device use by mineralogists and structural geologists to represent 3-dimensional information in two dimensions.

Mineralogists use a Wulff stereonet, which is constructed from a simple geometric recipe. Structural geologists use a SchmidtFile Size: 19KB. The Astronomical Triangle: Angular Size In astronomy we are limited by the fact that we are looking at the sky as a two-dimensional spherical shell in the sky with no immediate knowledge of the distance to an object, Hence the true size or width across the sky of an astrophysical object in physical units of distance like km or light-years is.

Maths - Stereographic Projection - Riemann Sphere. This page overlaps with the page here, I need to combine them. The book explains how to represent complex transformations such as the Möbius transformations. It also shows how complex. This is a perspective projection on a plane tangent at the center point from the point antipodal to the center point.

The center point is a pole in the common polar aspect, but can be any point. This projection has two significant properties. It is conformal, being free from angular distortion. Astronomical Coordinate Systems The coordinate systems of astronomical importance are nearly all spherical coordinate systems.

The obvious reason for this is that most all astronomical objects are remote from the earth and so appear to move on the backdrop of the celestial sphere. While one may still use a spherical coordinateFile Size: KB.

Stereographic Projection from Four-Space. We have described features of stereographic projection from the sphere in three-space to a plane. To describe this technique in the next higher dimension, we consider the effect of central projection on the analogue of a sphere in four-dimensional space, which we call a hypersphere.

astronomical triangle A spherical triangle on the celestial sphere formed by the intersection of the great circles joining a celestial body, S, the observer's zenith, Z, and the north (or south) celestial pole, P (see illustration). The relationships between the angles and sides of a spherical triangle are used for transformation between equatorial and horizontal coordinate systems: the angle.

To get the coordinates of the point $Q$ from those of the point $P$ consider the two similar triangles (on the right). By comparing the vertical sides of these. Make a note explaining why must be the image of a great circle (using the properties of stereographic projection).

Make a GreatCircle AB tool. (This will have the equatorial givens as givens in the tool as well as A and B.) Use your tool to draw great circles that form a triangle ABC in S-geometry.

Note the triangle A*B*C*. Theorem 2: Stereographic projection is circle preserving. Proof: Pick a circle on S not containing N and let A be the vertex of the cone tangent to S at this circle (Fig.

In the plane NZAconstruct Ap parallel to traverses the circle, jA−Zj is constant, but Az and az make equal angles with Nz, so the triangle AZp is isoceles and jA File Size: 56KB. Stereographic Projection.

Stereographic projection is a way of projecting onto. Intuitively, it can be thought of as the process of puncturing the sphere and stretching it out flat, onto Euclidean space.

It too can be used as an aid to visualization. Stereographic Projection of. image in the projection sphere. Some more words: rolling circles. The sphere that we have chosen for our stereographic projection only has some very identified details (equator, poles, polar diameter), reason why it shines a transparent point to our curious eyes.

There is a curve that deserves a special mention: the cycloid, the hypocycloid and the. The scale of the stereographic projection increases with distance from the point of tangency, but it increases more slowly than in the gnomonic projection.

The stereo-graphic projection can show an entire hemisphere without excessive distortion (Figure b).As in other azimuthal projections, great circles through the point of tangency appear as straight lines.

Stereographic projection has an important quality that each spatial circle is represented by another circle in the plane.

Construction of the main circles according to Křišťan. The dial of the Prague Astronomical Clock uses a projection from the North Pole. The centre of the dial represents the South Pole. Under stereographic projection, this circle is transformed to a circle on the plane, which is also considered as the stereographic image of the point outside the sphere onto the plane.

The coordinates of a point of the three-dimensional space are considered as tetracyclic coordinates of the circle on the plane. Geometry - Stereographic projection Thread starter Pearce_09; Start date ; #1 Pearce_ 74 0. Main Question or Discussion Point. I know if a cirlce (on S^2) does not contain N (0,0,1) then it is mapped onto the plane H as a circle.

Now say the circles on S^2 are lines of latitude. The Riemann sphere can be visualized as the unit sphere x2 + y2 + z2 = 1 in the three-dimensional real space R3. To this end, consider the stereographic projection from the unit sphere minus the point (0, 0, 1) onto the plane z = 0, which we identify with the complex plane by ζ = x + iy.

In Cartesian coordinates (x, y, z) and. Lab 3 Introduction to Stereographic Projection In this experiment, the aim is to provide a practical and theoretical introduction to the stereographic projection in order to use it in morphological crystallography of polycrystalline materials.

The stereographic projection is a projection of points from the surface of a sphere on to its File Size: KB. stereographic projection[¦sterēə¦grafik prə′jekshən] (crystallography) A method of displaying the positions of the poles of a crystal in which poles are projected through the equatorial plane of the reference sphere by lines joining them with the south pole for poles in the upper hemisphere, and with the north pole for poles in the lower.

where the equator falls at inﬁnity!). The advantage: straight lines in the projection are great circles on the sphere, so it’s really good for airplane pilots. Stereographic: The focal point is at the opposite “pole” of the sphere.

This is a conformal projection and is therefore highly desireable for astronomy. For a hemisphere, the File Size: KB. The projection point for the stereographic projection is the North pole; on astrolabes the South pole is more common.

The ecliptic dial makes one complete revolution in 23 hours 56 minutes (a sidereal day), and will therefore gradually get out of phase with the hour hand, drifting slowly further apart during the year.

Stereographic projection is one way to make a flat map of the earth. Because the earth is spherical, any map must distort shapes or sizes to some degree. Mathematically, a projection (or type of map) is described by a rule telling where (in the plane of the map paper) to draw the image of each point on the sphere.

The Astrolabium Catholicum. P-Al parallactic triangle performed by rotating Plancius point of departure Regular astronomical astrolabe represents a stereographic rete and limbus RIGHT ASCENSION Rijksmuseum rotating ruler rotating the ruler ruler and arm slide with arm sphere in stereographic spherical star's longitude stereographic.Stereographic projection maps circles of the unit sphere, which contain the north pole, to Euclidean straight lines in the complex plane; it maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane.

Proof. Let a circle c on the unit sphere Σ be given. Then this circle c is the set of all.The stereographic projection allows complex spherical trigonometry problems to be solved graphically.

For a planispheric astrolabe, the projection plane is the equatorwith the observer at the south s on the celestial sphere are projected as circles on the projection planeAngles between objects on Size: 3MB.