Subtended
Encyclopedia
In geometry
, an angle
subtended by an arc
, line or other curve is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context. For example, one may speak of the angle subtended by an arc of a circumference
when the angle's vertex
is a point on the circumference. A simple theorem of plane geometry states that arcs of equal lengths subtend equal angles in such a situation.
Similarly, a surface or solid object subtends a solid angle. A solid angle
is the area
on a unit sphere
cut out by the envelope of the vectors defining the perimeter
of the surface or object. The solid angle subtended by a circular cone
of opening angle (aperture) θ is
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, an angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
subtended by an arc
Arc (geometry)
In geometry, an arc is a closed segment of a differentiable curve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle...
, line or other curve is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context. For example, one may speak of the angle subtended by an arc of a circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
when the angle's vertex
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
is a point on the circumference. A simple theorem of plane geometry states that arcs of equal lengths subtend equal angles in such a situation.
Similarly, a surface or solid object subtends a solid angle. A solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
is the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
on a unit sphere
Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
cut out by the envelope of the vectors defining the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
of the surface or object. The solid angle subtended by a circular cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
of opening angle (aperture) θ is
External links
- How an object subtends an angle, Math Open Reference, With interactive appletJava appletA Java applet is an applet delivered to users in the form of Java bytecode. Java applets can run in a Web browser using a Java Virtual Machine , or in Sun's AppletViewer, a stand-alone tool for testing applets...
- Angle definition pages, Math Open Reference, with interactive applets that are also useful in a classroom setting.