# Track point

Trace point is a term used in analytical and representational geometry , which refers to the intersection of lines and planes in three-dimensional space with the coordinate planes or axes . ${\ displaystyle \ mathbb {R} ^ {3}}$

## Track points of a straight line

Track points of a straight line

The points of intersection of the straight line with the coordinate planes are referred to as the track points of a straight line in three-dimensional space . The point at which the straight line penetrates the xy basic plane with the equation means that the track points and are defined analogously . For example, if a straight line equation is given in parametric form as follows ${\ displaystyle \ mathbb {R} ^ {3}}$${\ displaystyle z = 0}$${\ displaystyle S_ {xy}}$${\ displaystyle S_ {xz}}$${\ displaystyle S_ {yz}}$

${\ displaystyle {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}} = {\ begin {pmatrix} 3 \\ 2 \\ - 4 \ end {pmatrix}} + \ lambda {\ begin { pmatrix} 4 \\ 3 \\ - 2 \ end {pmatrix}}}$with ,${\ displaystyle \ lambda \ in \ mathbb {R}}$

then the results from zeroing component: . The position vector of the tracking point is by inserting determined in the parametric representation: . The track point therefore has the coordinates . ${\ displaystyle z}$${\ displaystyle \ lambda = -2}$${\ displaystyle \ lambda}$${\ displaystyle {\ vec {s}} _ {xy} = {\ begin {pmatrix} -5 \\ - 4 \\ 0 \ end {pmatrix}}}$${\ displaystyle S_ {xy} = P (-5 \ mid -4 \ mid 0)}$

The prerequisite for the existence of a track point with a coordinate plane is that the straight line must not run parallel to this plane.

## Track points of a plane

The track points of a plane in three-dimensional space are the points of intersection of the plane with the coordinate axes. Their designation is based on the coordinate axis that is intersected in each case. The calculation can be made from the intercept form or the coordinate form of a plane equation . ${\ displaystyle \ mathbb {R} ^ {3}}$

For example, if the plane is given as follows in coordinate form: so is obtained by zeroing the - and component: . The track point thus has the coordinates . The two further track points can be determined accordingly. ${\ displaystyle 6x + 4y-3z = 12}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle x = 2}$${\ displaystyle S_ {x} = P (2 \ mid 0 \ mid 0)}$

The prerequisite for the existence of a track point with one of the coordinate axes is that it must not run parallel to one of the coordinate planes.