Stereographic Projections

1. Stereographic Projections Jacqueline Hess

2. History • Earliest known uses were in Greece for map making • Earliest references in literature (Roman, ~100 B.C.) • Vitruvius ~ Ten Books on Architecture • Ptolemy’s ~ Representation of the Sphere in the plane • Stereographic projection was used for a horoscopic instrument for determining time • Horoscope ~ the point of intersection of the ecliptic & eastern part of the horizon determined by means of this instrument • The ancients knew how to prove • Circles through the pole are mapped onto straight lines • Other circles are mapped onto circles • “Stereographic projection” • Introduced by Francois D’Aguillon (1566-1617) ~ Six Books of Optics • 10th Century Scholar al-Saghani suggested Gnomic Projection • Projection form an arbitrary point on the axis – if that point is the center of the sphere

3. Stereographic Projections • How projected • It is projected from a center point • Advantages • All circles on sphere plot as circles on plane • Drawbacks • Radial Distortion • Uses • Most widely used projection in mineralogy and structural geology

4. Stereographic Projections • “Family of Projections” that map from the sphere directly onto the plane • Maps everything except (0,0,1) onto whole plane z = -1

5. Stereographic Projections

6. Stereographic Projections

7. Orthographic Projections • How projected • From sphere perpendicular to plane • Advantages • True visual view • All circles plot as ellipses or straight lines • Drawbacks • Distortion near edges • Uses • Mostly used in structural geology for drawing block diagrams

8. Orthographic Projections • Think Perpendicular • Maps southern hemisphere onto unit circle

9. Orthographic Projections • It is a parallel projection that shows the Earth as it would be viewed from deep space. The parallel projection lines are perpendicular to the projection plane. In the Polar Orthographic projection, the projection plane is tangent to the earth's pole (see diagram at right).The Orthographic projection has become popular since the first photographs were taken in space in the 1960s, and it is excellent for showing what the earth looks like from outer space. But it is not widely used to display scientific data because distortion is very high near the edges, and only half the earth can be shown. • Polar Orthographic projection. The projection plane is tangent to pole.

10. Orthographic Projections

11. Gnomic Projections • How projected • From center of sphere • Advantages • Great circles always plot as straight lines • Drawbacks • Radial distortion • Uses • Mineralogy

12. Gnomic Projections • Center is projected out radially • Only 1 hemisphere • Maps southern hemisphere were plane z = -1

13. Gnomic Projections

14. Gnomic Projections It is projected from the center of the earth. It is also called the “Central Projection,” or simply “Gnomic.” This projection displays less than half of the earth: The Gnomonic projection is very distorted in all but one respect: Great Circles are always displayed as straight lines. That's right, any straight line drawn on a Gnomonic map will be the shortest distance between two points. The scale will be badly distorted along the line, but the route will be precise for the sphere.

15. Equal Area Projections • How projected • Draw an arc from point on sphere to plane • Advantages • Area conserved & moderate distortion • Drawbacks • Curves are complex • Uses • Structural geology • For statistical analysis of spatial data

16. Equal Area Projections • Arc length lies flat

17. http://www.3dsoftware.com/Cartography/USGS/MapProjections/ http://www.uwgb.edu/dutchs/structge/sphproj.htm http://www.technion.ac.il/guides/matlab/toolbox/map/proj43.html Citations