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(the left side of the equation uses the Hesse normal form of a line to calculate the distance | |). For a parametric equation of a parabola in general position see § As the affine image of the unit parabola.
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector of length one is called a unit normal vector.
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue. In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane, a plane in Euclidean space, or a hyperplane in higher dimensions.
This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by: + =, where is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x-axis to this segment), and p is the (positive) length of the normal ...
A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −(k − t)x/t + k − t. Rearranging and casting in the form F(x,y,t) = 0 gives:
The equation of a line is given by = +. The equation of the normal of that line which passes through the point P is given = +. The point at which these two lines intersect is the closest point on the original line to the point P. Hence:
The circle S and the curve C have the common tangent line at P, and therefore the common normal line. Close to P, the distance between the points of the curve C and the circle S in the normal direction decays as the cube or a higher power of the distance to P in the tangential direction.
A particular tangent line distinguishes the hyperbola from the other conic sections. [18] Let f be the distance from the vertex V (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2f. The tangent ...