Set of pyramidal frusta | |
---|---|
Examples: Pentagonal and square frusta | |
Faces | n trapezoids, 2 n-gons |
Edges | 3n |
Vertices | 2n |
Symmetry group | C_{nv} |
Dual polyhedron | - |
Properties | convex |
- For the graphics technique known as Frustum culling, see Hidden surface determination
A frustum ^{[1]} (plural: frusta or frustums) is the portion of a solid—normally a cone or pyramid—which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen (which is formed by a clipped pyramid).
Each plane section is a floor of the frustum. The axis of the frustum, if any, is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.
Cones and pyramids can be viewed as degenerate cases of frustums, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoids.
Two frusta joined at their bases make a bifrustum.
Formulas
Volume
The volume formula of frustum of square pyramid was introduced by the ancient Egyptian mathematics in what is called the Moscow Mathematical Papyrus, written ca. 1850 BC.:
- $ V = \frac{1}{3} h(a^2 + a b +b^2). $
where a and b are the base and top side lengths of the truncated pyramid, and h is the height. The Egyptians knew the correct formula for obtaining the volume of a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus.
The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:
- $ V = \frac{h_1 B_1 - h_2 B_2}{3} $
where B_{1} is the area of one base, B_{2} is the area of the other base, and h_{1}, h_{2} are the perpendicular heights from the apex to the planes of the two bases.
Considering that
- $ \frac{B_1}{h_1^2}=\frac{B_2}{h_2^2}=\frac{\sqrt{B_1 B_2}}{h_1 h_2} = $ α
the formula for the volume can be expressed as a product of this proportionality α/3 and a difference of cubes of heights h_{1} and h_{2} only.
By factoring (h_{2}−h_{1}) = h, the height of the frustum, and α(h_{1}^{2} + h_{1}h_{2} + h_{2}^{2})/3, and distributing α and substituting from its definition, the Heronian mean of areas B_{1} and B_{2} is obtained. The alternative formula is therefore
- $ V = \frac{h}{3}(B_1+\sqrt{B_1 B_2}+B_2) $
Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number, the square root of negative one.^{[2]}
In particular, the volume of a circular cone frustum is
- $ V = \frac{\pi h}{3}(R_1^2+R_1 R_2+R_2^2) $
where π is 3.14159265..., and R_{1}, R_{2} are the radii of the two bases.
The volume of a pyramidal frustum whose bases are n-sided regular polygons is
- $ V= \frac{n h}{12} (a_1^2+a_1a_2+a_2^2)\cot \frac{\pi}{n} $
where a_{1} and a_{2} are the sides of the two bases.
Surface area
For a right circular conical frustum^{[3]}
- $ \begin{align}\text{Lateral Surface Area}&=\pi(R_1+R_2)s\\ &=\pi(R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}\end{align} $
and
- $ \begin{align}\text{Total Surface Area}&=\pi((R_1+R_2)s+R_1^2+R_2^2)\\ &=\pi((R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}+R_1^2+R_2^2)\end{align} $
where R_{1} and R_{2} are the base and top radii respectively, and s is the slant height of the frustum.
The surface area of a right frustum whose bases are similar regular n-sided polygons is
- $ A= \frac{n}{4}\left[(a_1^2+a_2^2)\cot \frac{\pi}{n} + \sqrt{(a_1^2-a_2^2)^2\sec^2 \frac{\pi}{n}+4 h^2(a_1+a_2)^2} \right] $
where a_{1} and a_{2} are the sides of the two bases.
Examples
- An example of a pyramidal frustum may be seen on the reverse of the Great Seal of the United States, as on the back of the U.S. one-dollar bill. The "unfinished pyramid" is surmounted by the "Eye of Providence".
- Certain ancient Native American mounds also form the frustum of a pyramid.
- The John Hancock Center in Chicago, Illinois is a frustum whose bases are rectangles.
- The Washington Monument is a narrow pyramidal frustum (with square bases) with a pyramid attached to the top base.
- In 3D computer graphics, the usable field of view of a virtual photographic or video camera is modeled as a pyramidal frustum, the viewing frustum.
- The cone shaped fairing which forms the transition between two stages of a multistage rocket, when the stages are of different diameters (see Saturn V, Delta III, Ares I), is an example of a conic frustum. In the industry, the term 'frustum' is used to refer to this piece.
Note
- ↑ frustum is Latin and means piece, crumb. The English word is often misspelled as Template:Sic, probably because of a similarity with the common words frustrate and frustration, also of Latin origin.
- ↑ Nahin, Paul. "An Imaginary Tale: The story of [the square root of minus one]." Princeton University Press. 1998
- ↑ "Mathwords.com: Frustum". http://www.mathwords.com/f/frustum.htm. Retrieved 17 July 2011.
External links
Template:Wiktionarypar Template:Commonscat
- Weisstein, Eric W., "Pyramidal frustum" from MathWorld.
- Weisstein, Eric W., "Conical frustum" from MathWorld.
- Paper models of frustums (truncated pyramids)
- Paper model of frustum (truncated cone)
- Design paper models of conical frustum (truncated cones)cs:Komolý jehlan
da:Keglestubeo:Trunko (geometrio)it:Tronco (geometria) nl:Afgeknotte piramide pt:Tronco de bases paralelas th:ฟรัสตัม sv:Avstympat parti