Little-Fish

指導教授：全任重教授

National
Tsing Hua University

=國立清華大學=

[幾何作圖--期末作品呈現]

Final Project Presentation for
Geometric Construction

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About 作者

Cabri3D -作品集

軟體資訊

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可以進入頁面看cabri 3D立體圖形

幾何‧作品

[卡塔蘭立體-半正多面體-對偶多面體] Go To See

[卡塔蘭立體]:

卡塔蘭立體是半正多面體的對偶多面體，都是凸多面體。1865年比利時數學家歐仁·查理·卡塔蘭最先描述它們。
卡塔蘭立體面勻稱而點不勻稱，而其對偶多面體半正多面體點勻稱而面不勻稱。只有兩個邊均稱的卡塔蘭立體：菱形十二面體和菱形三十面體。目前共計有13種卡塔蘭立體，其對偶多面體均為半正多面體(阿基米德立體)

[半正多面體]:

半正多面體是泛指所有由2種以上正多邊形所組成的多面體

[對偶多面體]:

在幾何學，若一種多面體的每個頂點均能對應到另一種多面體上的每個面的中心，它就是對方的對偶多面體。
根據對偶原則，每種多面體都存在對偶多面體。一種多面體的對偶多面體的對偶多面體等同該種多面體。

[有關菱形]: Go To See

許多菱形多面體的東西，那些名詞不是數學家發明的，而是來自礦物學!現在我們所做的許多東西，是為了模擬那些礦物，而產生的模型。

[組合]: Go To See

把各種不同地多面體塞在一起，研究旗美麗的數學性質。

[趣味]: Go To See

生活周遭的圖形。

[截半立方體]:Go To See

截半立方體是立方體和其對偶多面體正八面體的立體混合物，是半正多面體的一種。它是用立方體由一條棱斬到另一條棱的中點（即斬去立方體的頂點）而成。
截半立方體每六條棱可以成為一個正六邊形，共有四個獨立的六邊形。
從中切一半的話，叫三角平面塔Triangular Cupola

[相同立體旋轉組合]: Go To See

有相同中心的立方體

[圓錐曲線]: Go To See

將高中數學立體化，概念更清楚。

[縮放動畫]: Go To See

圖形的縮放造成美麗的性質。

[有關正四面體的組合]: Go To See

正四面體的拼湊與旋轉。

[Poncelet's Porism 與 Steiner Porism]: Go To See

Poncelet's Porism

In geometry, Poncelet's porism (sometimes referred to as Poncelet's closure theorem), named after French engineer and mathematician Jean-Victor Poncelet, states the following: Let C and D be two plane conics. If it is possible to find, for a given n > 2, one n-sided polygon that is simultaneously inscribed in C and circumscribed around D (i.e., a bicentric polygon), then it is possible to find infinitely many of them.

Steiner Porism

In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles, where n is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual closed Steiner chains, the first and last (nth) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles α and β do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.
Steiner chains are named after Jakob Steiner, who defined them in the 19th century and discovered many of their properties.

[Villarceau circle & 軌跡]: Go To See

Villarceau circle

In geometry, Villarceau circles are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane (containing the point) parallel to the equatorial plane of the torus. Another is perpendicular to it. The other two are Villarceau circles.
They are named after the French astronomer and mathematician Yvon Villarceau (1813–1883).

[Pappus theorem]: Go To See

Pappus theorem = Pappus's hexagon theorem

In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes.

[反演技巧]: Go To See

反演

反演是種幾何變換。給定點O、常數k，點P的變換對應點就是在以O開始的射線OP上的一點 P'使得|OP||OP' | = k2。
反演的結果：
→過點O的直線：直線
→過點O的圓：不過點O的直線
→不過點O的圓：圓
→過點O的球：不過點O的平面

[投影-多面體]: Go To See
[投影part 2-三角錐]: Go To See
[投影part 3-球]: Go To See

參考‧資料

數學家歷史 http://www-history.mcs.st-and.ac.uk/
wiki
Wolfram MathWorld http://mathworld.wolfram.com/