Motion planning in the context of "Line–line intersection"

In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán.

Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically. Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. Kalman filtering is also important for robotic motion planning and control, and can be used for trajectory optimization. Kalman filtering also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, the use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands.

In Euclidean geometry, the intersection of a line and a line can be the empty set, a single point, or a line (if they coincide). Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection.

In a Euclidean space, if two lines are not coplanar, they have no point of intersection and are called skew lines. If they are coplanar, however, there are three possibilities: if they coincide (are the same line), they have all of their infinitely many points in common; if they are distinct but have the same direction, they are said to be parallel and have no points in common; otherwise, they have a single point of intersection, denoted as singleton set, for instance ${\displaystyle \{A\}}$.

In geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or the line itself. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.

Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in computer graphics, motion planning, and collision detection.