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For the graphics technique known as Frustum culling, see Hidden surface determination
Hidden surface determination

In 3D computer graphics, hidden surface determination is the process used to determine which surfaces and parts of surfaces are not visible from a certain viewpoint....
A frustum (plural: frusta or frustums) is the portion of a solid
Solid

A solid object is in the states of matter characterized by resistance to deformation and changes of volume. In other words, it has high values both of Young's modulus and of shear modulus; this contrasts e.g....
—normally a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 or pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
—which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
 to describe the 3d area which is visible on the screen (which is formed by a clipped pyramid).

Elements, special cases, and related concepts
Each plane section is a base of the frustum.






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For the graphics technique known as Frustum culling, see Hidden surface determination
Hidden surface determination

In 3D computer graphics, hidden surface determination is the process used to determine which surfaces and parts of surfaces are not visible from a certain viewpoint....
A frustum (plural: frusta or frustums) is the portion of a solid
Solid

A solid object is in the states of matter characterized by resistance to deformation and changes of volume. In other words, it has high values both of Young's modulus and of shear modulus; this contrasts e.g....
—normally a cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 or pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
—which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics
Computer graphics

Computer graphics are graphics created by computers and, more generally, the representation and manipulation of pictorial data by a computer....
 to describe the 3d area which is visible on the screen (which is formed by a clipped pyramid).

Elements, special cases, and related concepts


Each plane section is a base 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
Apex (geometry)

In geometry, an apex is a descriptive label for a visual singular highest or most distant point or Vertex in an isosceles triangle, Pyramid or Cone , usually contrasting with the opposite side called the base....
 (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoid
Prismatoid

A prismatoid is a polyhedron where all vertices lie in two parallel planes. If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height is h, then the volume of the prismatoid i...
s.

Two frusta joined at their bases make a bifrustum
Bifrustum

An n-agonal bifrustum is a polyhedron composed of three parallel planes of polygon, with the middle plane largest and usually the top and bottom congruent....
.

Formulas


The volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 of a frustum is the difference between the volume of the cone (or other figure) before slicing the apex off, minus the volume of the cone (or other figure) that was sliced off: where and are the perpendicular heights from the apex to the planes of the smaller and larger base, , are the areas of the two bases.

Let be the height of the frustum, that is, the perpendicular distance between the two planes. Considering that and , one gets the alternative formula for the volume

(See Heronian mean
Heronian mean

The Heronian mean H of two non-negative real numbers A and B is given by the formula:It is named after Hero of Alexandria, and used in finding the volume of a frustum of a pyramid or cone ....
.)

In particular, the volume of a circular cone
Cone (geometry)

A cone is a dimension geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base....
 frustum is where
Pi

Pi or p is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean geometry; this is the same value as the ratio of a circle's area to the square of its radius....
 is 3.14159265..., and , are the radii 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
    Great Seal of the United States

    The Great Seal of the United States is used to authenticate certain documents issued by the Federal government of the United States. The phrase is used both for the physical seal itself , and more generally for the design impressed upon it....
    , as on the back of the U.S. one-dollar bill. The "unfinished pyramid" is surmounted by the "Eye of Providence
    Eye of Providence

    The Eye of Providence, or the all-seeing eye, is a symbol showing an eye surrounded by Ray of light or a Glory #Glory in Art and usually enclosed by a triangle....
    ".
  • Certain ancient Native American
    Indigenous peoples of the Americas

    The indigenous peoples of the Americas are the pre-Columbian inhabitants of the Americas, their descendants, and many ethnic groups who identify with those peoples....
     mounds also form the frustum of a pyramid.
  • The John Hancock Center
    John Hancock Center

    John Hancock Center at 875 North Michigan Avenue in the Streeterville area of the Near North Side, Chicago in Chicago, Illinois, is a 100-Storey, 1,127-foot tall skyscraper designed by structural engineer Fazlur Khan of Skidmore, Owings and Merrill....
     in Chicago
    Chicago

    Chicago is the largest city in the U.S. state of Illinois and the Midwestern United States, as well as the List of United States cities by population city in the United States with more than 2.8 million residents....
    , Illinois
    Illinois

    The State of Illinois is a U.S. state of the United States, the 21st to be admitted to the United States. Illinois is the most populous and demographically diverse Midwestern United States state and the fifth most populous state in the nation....
     is a frustum whose bases are rectangles.
  • The Washington Monument
    Washington Monument

    The Washington Monument is a large, tall, sand-colored obelisk near the west end of the National Mall in Washington, D.C. It is a United States Presidential Memorial constructed to commemorate the first U.S....
     is a narrow pyramidal frustum (with square bases) with a pyramid attached to the top base.
  • In 3D computer graphics
    3D computer graphics

    3D computer graphics are graphics that use a Cartesian coordinate system#Three-dimensional coordinate system representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images....
    , the usable field of view of a virtual photographic or video camera
    Camera

    A camera is a device that records images, either as a still photograph or as moving images known as videos or movies. The term comes from the camera obscura , an early mechanism of projecting images where an entire room functioned as a real-time imaging system; the modern camera evolved from the camera obscura....
     is modeled as a pyramidal frustum, the viewing frustum
    Viewing frustum

    In 3D computer graphics, the viewing frustum or view frustum is the region of space in the modeled world that may appear on the screen; it is the field of view of the notional camera....
    .
  • The cone shaped fairing which forms the transition between two stages of a multistage rocket
    Multistage rocket

    A multistage rocket is a rocket that usestwo or more stages, each of which contains its own Rocket engine and Rocket propellant. A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage....
    , when the stages are of different diameters (see Saturn V
    Saturn V

    The Saturn V was a multistage rocket liquid-fuel expendable launch system rocket used by NASA's Apollo program and Skylab programs from 1967 until 1973....
    , Delta III, Ares I
    Ares I

    Ares I is the crew launch vehicle being developed by NASA as a component of Constellation program. NASA plans to use Ares I to launch Orion , the spacecraft being designed for NASA human spaceflight missions after the Space Shuttle is retired in 2010....
    ), is an example of a conic frustum. In the industry, the term 'frustum' is used to refer to this piece.


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