First Six Books of the Elements of Euclid

... A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness. ...

... A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness. ...

geom practice worksheet answers

... 3. The converse, “If ∆n∆ = 17, then n = 17,” is not true. 4. A figure has eight sides if and only if it is an octagon. 5. If a whole number is a multiple of 5, then its last digit is either 0 or 5. If a whole number has a last digit of 0 or 5, then it is a multiple of 5. 6. If two lines are perpendi ...

... 3. The converse, “If ∆n∆ = 17, then n = 17,” is not true. 4. A figure has eight sides if and only if it is an octagon. 5. If a whole number is a multiple of 5, then its last digit is either 0 or 5. If a whole number has a last digit of 0 or 5, then it is a multiple of 5. 6. If two lines are perpendi ...

CK-12 Trigonometry

... The variable x is often referred to as the independent variable, while the variable y is referred to as the dependent variable. We talk about x and y this way because the y values of a function depend on what the x values are. That is why we also say that “y is a function of x.” For example, the val ...

... The variable x is often referred to as the independent variable, while the variable y is referred to as the dependent variable. We talk about x and y this way because the y values of a function depend on what the x values are. That is why we also say that “y is a function of x.” For example, the val ...

Content, Methods, and Context of the Theory of Parallels

... within a decade of his death, the publication of his correspondence drew the attention of the mathematical world to non-Euclidean geometry. Though the notion that there could be two geometries did indeed generate controversy, the fact that Gauss himself endorsed it was enough to convince several mat ...

... within a decade of his death, the publication of his correspondence drew the attention of the mathematical world to non-Euclidean geometry. Though the notion that there could be two geometries did indeed generate controversy, the fact that Gauss himself endorsed it was enough to convince several mat ...

# 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.