Theorems powerpoint
... 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four rig ...
... 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four rig ...
Geometry
... If a line is tangent to a circle, then the line is to the radius drawn to the point of tangency. If a line is to a radius at its outer endpoint, then the line is tangent to the circle. Minor arcs are if and only if their central angles are . In the same circle or in circles: (1) ...
... If a line is tangent to a circle, then the line is to the radius drawn to the point of tangency. If a line is to a radius at its outer endpoint, then the line is tangent to the circle. Minor arcs are if and only if their central angles are . In the same circle or in circles: (1) ...
Geometry Glossary Essay, Research Paper Geometry Glossary
... - two rays with a common endpoint that form a line Ordered pair - the two numbers that (called coordinates) are used to identify a point in a plane; written (x, y) Ordered triple - the three numbers (called coordinates) that are used to identify a point in space; written (x, y, z) Orientation - in a ...
... - two rays with a common endpoint that form a line Ordered pair - the two numbers that (called coordinates) are used to identify a point in a plane; written (x, y) Ordered triple - the three numbers (called coordinates) that are used to identify a point in space; written (x, y, z) Orientation - in a ...
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called ""Euler angles"". In that case, the sequences of the first group are called proper or classic Euler angles.