half-plane

(noun)

One of the two parts of the coordinate plane created when a line is drawn.

Related Terms

  • boundary line

Examples of half-plane in the following topics:

  • Graphing Inequalities

    • A straight line drawn through the plane divides the plane into two half-planes, as shown in the diagram below.
    • Next, choose a test point to figure out which half-plane we need to shade in.
    • This a true statement, so shade in the half-plane containing $(0, 0). $
    • All points in the shaded half-plane above the line are solutions to this inequality.
    • The boundary line shown above divides the coordinate plane into two half-planes.
  • Animal Body Planes and Cavities

    • A sagittal plane divides the body into right and left portions.
    • A frontal plane (also called a coronal plane) separates the front (ventral) from the back (dorsal).
    • A transverse plane (or, horizontal plane) divides the animal into upper and lower portions.
    • The midsagittal plane divides the body exactly in half into right and left portions.
    • The frontal plane divides the front and back, while the transverse plane divides the body into upper and lower portions.
  • The Cartesian System

    • Named for "the father of analytic geometry," 17th-century French mathematician René Descartes, the Cartesian coordinate system is a 2-dimensional plane with a horizontal axis and a vertical axis.
    • Points are specified uniquely in the Cartesian plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes.
    • Therefore, you move one and a half units left and two and a half units down.
    • A Cartesian plane is particularly useful for plotting a series of points that show a relationship between two variables.
  • Body Planes and Sections

    • There are three basic reference planes used in anatomy: the sagittal plane, the coronal plane, and the transverse plane.
    • Body planes are hypothetical geometric planes used to divide the body into sections.
    • Reference planes are the standard planes used in anatomical terminology and include:
    • A longitudinal plane is any plane perpendicular to the transverse plane, while parasaggital planes are parallel to the saggital plane.
    • The coronal plane, the sagittal plane, and the parasaggital planes are examples of longitudinal planes.
  • Conic Sections in Polar Coordinates

    • In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane.
    • One of the most useful definitions, in that it involves only the plane, is that a conic consists of those points whose distances to some point—called a focus—and some line—called a directrix—are in a fixed ratio, called the eccentricity.
    • where e is the eccentricity and l is half the latus rectum.
  • Water Waves

    • In the case of monochromatic linear plane waves in deep water, particles near the surface move in circular paths, creating a combination of longitudinal (back and forth) and transverse (up and down) wave motions.
    • When waves propagate in shallow water (where the depth is less than half the wavelength), the particle trajectories are compressed into ellipses.
    • Deep water corresponds with a water depth larger than half the wavelength, as is a common case in the sea and ocean.
    • The deep-water group velocity is half the phase velocity.
    • The wave can be thought to be made up of planes orthogonal to the direction of the phase velocity.
  • Conditions for Wave Interference: Reflection due to Phase Change

    • A simple form of wave interference is observed when two waves of the same frequency (also called a plane wave) intersect at an angle , as shown in .
    • Destructive interference occurs when the waves are half a cycle out of phase, or
    • In other words, the wave undergoes a 180 degree change of phase upon reflection, and the reflected ray "jumps" ahead by half a wavelength.
  • Spherical and Plane Waves

    • Spherical waves come from point source in a spherical pattern; plane waves are infinite parallel planes normal to the phase velocity vector.
    • A plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector .
    • It is not possible in practice to have a true plane wave; only a plane wave of infinite extent will propagate as a plane wave.
    • However, many waves are approximately plane waves in a localized region of space.
    • Plane waves are an infinite number of wavefronts normal to the direction of the propogation.
  • Vectors in the Plane

    • Vectors are needed in order to describe a plane and can give the direction of all dimensions in one vector equation.
    • Planes in a three dimensional space can be described mathematically using a point in the plane and a vector to indicate its "inclination".
    • As such, the equation that describes the plane is given by:
    • which we call the point-normal equation of the plane and is the general equation we use to describe the plane.
    • This plane may be described parametrically as the set of all points of the form$\mathbf R = \mathbf {R}_0 + s \mathbf{V} + t \mathbf{W}$, where $s$ and $t$ range over all real numbers, $\mathbf{V}$ and $\mathbf{W}$ are given linearly independent vectors defining the plane, and $\mathbf{R_0}$ is the vector representing the position of an arbitrary (but fixed) point on the plane.
  • Equations of Lines and Planes

    • A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.
    • A line is essentially a representation of a cross section of a plane, or a two dimensional version of a plane which is a three dimensional object.
    • The components of equations of lines and planes are as follows:
    • This direction is described by a vector, $\mathbf{v}$, which is parallel to plane and $P$ is the arbitrary point on plane $M$.
    • where $t$ represents the location of vector $\mathbf{r}$ on plane $M$.
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