plane

Examples of plane in the following topics:

• 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.

• 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 $M$.

• Tangent Planes and Linear Approximations

  • The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
  • Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
  • Note the similarity of the equations for tangent line and tangent plane.
  • Since a tangent plane is the best approximation of the surface near the point where the two meet, tangent plane can be used to approximate the surface near the point.
  • The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.

• 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.
  • Shown are the planes of a quadruped goat and a bipedal human.
  • The frontal plane divides the front and back, while the transverse plane divides the body into upper and lower portions.

• Graphing Inequalities

  • A straight line drawn through the plane divides the plane into two half-planes, as shown in the diagram below.
  • This a true statement, so shade the half-plane containing $(0, 0).$
  • The boundary line shown above divides the plane into two half-planes
  • All points lying on the line and in the shaded half-plane are solutions.
  • Graph an inequality by shading the correct section of the plane

• Inconsistent and Dependent Systems in Three Variables

  • Graphically, the solutions fall on a line or plane that is the intersection of three planes in space.
  • Notice that two of the planes are the same and they intersect the third plane on a line.
  • The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
  • (b) Two of the planes are parallel and intersect with the third plane, but not with each other.
  • Two equations represent the same plane, and these intersect the third plane on a line.

• Tangent Vectors and Normal Vectors

  • Not only can vectors be ‘normal' to objects, but planes can also be normal.
  • These vectors are normal to the plane because the intersection between them and the plane makes a right angle.
  • A plane can be determined as normal to the object if the directional vector of the plane makes a right angle with the object at its tangent point.
  • This plane is normal to the point on the sphere to which it is tangent.
  • Each point on the sphere will have a unique normal plane.

• Green's Theorem

  • Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
  • Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.
  • Let $C$ be a positively oriented, piecewise smooth, simple closed curve in a plane, and let $D$ be the region bounded by $C$.
  • In plane geometry and area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
  • Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the $xy$-plane.