球極平面投影(stereographic projection),在幾何學裏,是一種將圓球面投影至平面的映射。在構造地質學裏,稱為球面立體投影或球面投影。除了投影點以外,這投影在整個球面都有定義。在這定義域裏,這映射具有光滑性、雙射性和共形性。共形性的意思就是角度維持不變。但是,這映射不會維持距離不變,也不會維持面積不變;它不會維持圖案的距離與面積。
此條目需要擴充。 (2014年11月22日) |
直覺而言,球極平面投影是一種以平面來看球面的方法。使用這方法,在圖案品質方面,必須接受一些不可避免的妥協。因為圓球與平面出現於許多數學方面的問題和應用,球極平面投影也非常地常見。在各個領域,例如,複分析,地圖學,地質學,與攝影,球極平面投影都有廣泛的用處。實際上,球極平面投影經常是用電腦繪成,或者用手工直接繪在一種特別的繪圖紙,稱為烏爾夫網圖。
公式
球面球座標投影到平面極座標:
反解:
特別地,當 R = 0 時,φ = π
圖片
-
這三維透視圖可以顯示出球極平面投影圖的製作方法。
-
瑞士沃倫山球極平面投影圖,由120張照片合成。
參考文獻
- 湯姆·麥克·阿波斯托. Mathematical Analysis 2. Addison-Wesley. 1974.
- Brown, James and Churchill, Ruel. Complex variables and applications. New York: McGraw-Hill. 1989. ISBN 0070109052.
- German, Daniel; Burchill, L.; Duret-Lutz, A.; Pérez-Duarte, S.; Pérez-Duarte, E.; Sommers, J. Flattening the Viewable Sphere. "Proceedings of Computational Aesthetics 2007". Banff: Eurographics: 23–28. June 2007.
- Do Carmo, Manfredo P. Differential geometry of curves and surfaces. Englewood Cliffs, New Jersey: Prentice Hall. 1976. ISBN 0-13-212589-7.
- Elkins, James. Did Leonardo Develop a Theory of Curvilinear Perspective?: Together with Some Remarks on the 'Angle' and 'Distance' Axioms. Journal of the Warburg and Courtauld Institutes (The Warburg Institute). 1988, 51: 190–196 [2010-10-16]. doi:10.2307/751275. (原始內容存檔於2021-05-20).
- Oprea, John. Differential geometry and applications. Englewood Cliffs, New Jersey: Prentice Hall. 2003. ISBN 0130652466.
- Pedoe, Dan. Geometry. Dover. 1988. ISBN 0-486-65812-0.
- Shafarevich, Igor. Basic Algebraic Geometry I. Springer. 1995. ISBN 0387548122.
- Snyder, John P. Map Projections − A Working Manual, Professional Paper 1395. US Geological Survey. 1987.
- Snyder, John P. An Album of Map Projections, Professional Paper 1453. US Geological Survey. 1989.
- Snyder, John P. Flattening the Earth. University of Chicago. 1993. ISBN 0-226-76746-9.
- Spivak, Michael. A comprehensive introduction to differential geometry, Volume IV. Houston, Texas: Publish or Perish. 1999. ISBN 091409873X.
外部連結
- 埃里克·韋斯坦因. Stereographic projection. MathWorld.
- http://planetmath.org/encyclopedia/StereographicProjection.html (頁面存檔備份,存於互聯網檔案館)
- Table of examples and properties of all common projections (頁面存檔備份,存於互聯網檔案館), from radicalcartography.net
- three dimensional Java Applet (頁面存檔備份,存於互聯網檔案館)
- Stereographic Projection and Inversion (頁面存檔備份,存於互聯網檔案館) from cut-the-knot
- Examples of miniplanet panoramas, majority in UK
- Examples of miniplanet panoramas, majority in Czech Republic (頁面存檔備份,存於互聯網檔案館)
- Examples of miniplanet panoramas, majority in Poland (頁面存檔備份,存於互聯網檔案館)
- DoITPoMS Teaching and Learning Package- "The Stereographic Projection" (頁面存檔備份,存於互聯網檔案館)
Wikiwand in your browser!
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.