cpctc
... are building a miniature golf course at your town’s youth center. The design plan calls for the first hole to have two congruent triangular bumpers. Prove that the bumpers on the first hole, shown at the right, meet the conditions of the plan. ...
... are building a miniature golf course at your town’s youth center. The design plan calls for the first hole to have two congruent triangular bumpers. Prove that the bumpers on the first hole, shown at the right, meet the conditions of the plan. ...
Geometry Lecture Notes
... common, i.e. l intersects m iff there exists A such that A is on l and A is on m. In this situation we say that the lines l and m intersect at A. A set of lines, all of which contain a point A are said to be concurrent. Definition The points in a figure are collinear if there exists a line containin ...
... common, i.e. l intersects m iff there exists A such that A is on l and A is on m. In this situation we say that the lines l and m intersect at A. A set of lines, all of which contain a point A are said to be concurrent. Definition The points in a figure are collinear if there exists a line containin ...
9 Neutral Triangle Geometry
... triangle, not the extensions—as follows from the exterior angle theorem. Hence the Crossbar Theorem implies that any two altitude do intersect. The remaining details are left as an exercise. Here is stated what we have gathered up to this point. It is little, but more than nothing! ...
... triangle, not the extensions—as follows from the exterior angle theorem. Hence the Crossbar Theorem implies that any two altitude do intersect. The remaining details are left as an exercise. Here is stated what we have gathered up to this point. It is little, but more than nothing! ...
A Method For Establishing Certain Trigonometric Inequalities
... with ∆ = 0 if and only if ∆∗ = 0. From this and from Theorem 2.2, it follows that u, v and w are the cosines of the angles of a triangle if and only if (2.7) and (2.8) (or equivalently (2.7) and (2.9)) hold. Also, the discriminant of ∆∗ , as a polynomial in q, is 16(3 − 2s)3 . Therefore for (2.9) to ...
... with ∆ = 0 if and only if ∆∗ = 0. From this and from Theorem 2.2, it follows that u, v and w are the cosines of the angles of a triangle if and only if (2.7) and (2.8) (or equivalently (2.7) and (2.9)) hold. Also, the discriminant of ∆∗ , as a polynomial in q, is 16(3 − 2s)3 . Therefore for (2.9) to ...
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.