What is an orthogonal trajectory of a family of curves?

An Orthogonal Trajectory of a family of curves is a curve that intersects each curve in the family of curves at right angles. Two curves intersect at right angles if their tangents at that point intersect at right angles. That is if the product of their slopes at the point of intersection is −1.

What are the orthogonal trajectories to the family of curves with equations?

The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in original family.

What is meant by orthogonal trajectory?

orthogonal trajectory, family of curves that intersect another family of curves at right angles (orthogonal; see figure).

What is the orthogonal trajectories of the given family of straight lines?

By replacing with we see that the orthogonal trajectories for the family of straight lines are concentric circles (Figure ):

How do you find an orthogonal family?

Video quote: All right in this problem we are asked to find the orthogonal trajectories for the family of curves of x equals K times y squared where K is an arbitrary constant.

How do you find orthogonal trajectories?

Our method of finding the orthogonal trajectories of a given family of curves is therefore as follows: first, find the differential equation of the family; next, replace dy/dx by dx/dy to obtain the differential equation of the orthogonal trajectories; and finally, solve this new differential equation.

How do you show orthogonal curves?

Two curves are said to be orthogonal if their tangent lines are perpendicular at every point of intersection. Two families of curves are said to be orthogonal if every curve in one family is orthogonal to every curve in the other family.

How do you find the orthogonal trajectories of a polar curve?

The orthogonal trajectories will be found by solving dydx=2xyx2−y2.

What is orthogonal linear transformation?

In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product.

What do you mean by orthogonality?

1a : intersecting or lying at right angles In orthogonal cutting, the cutting edge is perpendicular to the direction of tool travel. b : having perpendicular slopes or tangents at the point of intersection orthogonal curves.

Are all orthogonal transformations rotations?

In three-dimensional space, every special orthogonal transformation is a rotation around an axis, while every non-special orthogonal transformation is the product of such a rotation and a reflection in a perpendicular plane.

What is orthogonal matrix with example?

A square matrix with real numbers or elements is said to be an orthogonal matrix, if its transpose is equal to its inverse matrix. Or we can say, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

What is meant by orthogonal matrix?

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

How do you write an orthogonal matrix?

We construct an orthogonal matrix in the following way. First, construct four random 4-vectors, v₁, v₂, v₃, v₄. Then apply the Gram-Schmidt process to these vectors to form an orthogonal set of vectors. Then normalize each vector in the set, and make these vectors the columns of A.

What is orthogonal in maths?

Orthogonal is commonly used in mathematics, geometry, statistics, and software engineering. Most generally, it’s used to describe things that have rectangular or right-angled elements. More technically, in the context of vectors and functions, orthogonal means “having a product equal to zero.”

What are orthogonal circles?

Orthogonal circles are orthogonal curves, i.e., they cut one another at right angles. By the Pythagorean theorem, two circles of radii and whose centers are a distance apart are orthogonal if. (1) Two circles with Cartesian equations.

What is orthogonal architecture?

In computer engineering, an orthogonal instruction set is an instruction set architecture where all instruction types can use all addressing modes. It is “orthogonal” in the sense that the instruction type and the addressing mode vary independently.

What is the meaning of orthogonal in physics?

We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero.

How is orthogonality of two signals defined?

Any two signals say 500Hz and 1000Hz (On a constraint that both frequencies are multiple of its fundamental here lets say 100Hz) ,when both are mixed the resultant wave obtained is said to be orthogonal. Meaning: Orthogonal means having exactly 90 degree shift between those 2 signals.

What is orthogonal signal generation?

The orthogonal signal generators (OSGs), used in single-phase PLLs, are generally based on various types of filters, and they need to operate robustly in relation to the grid voltage disturbances and frequency variations.

What is orthogonality in communication?

Orthogonality means both signal is having phase difference of 90 degree. Hence, it will not interfere each other. Just like CDMA, all the channels are orthogonal and hence we can use same frequency allocation for all users but signals are decoded based on PN sequence which is used for spreading the signal.

Why orthogonality is important in communication?

Orthogonal signals are used extensively in communications because they can be received and demodulated as separate data streams with very little interference between the orthogonal signals.

What is the orthogonality thesis?

The Orthogonality Thesis states that an artificial intelligence can have any combination of intelligence level and goal, that is, its Utility Functions(107) and General Intelligence(65) can vary independently of each other.

What are the techniques used to maintain orthogonality in a system?

There are several techniques you can use to maintain orthogonality: Keep your code decoupled. Write shy code ”modules that don’t reveal anything unnecessary to other modules and that don’t rely on other modules’ implementations. Try the Law of Demeter [LH89], which we discuss in Decoupling and the Law of Demeter.