We can define distance from a point P to a plane α as the smallest of the distances from the point P to the infinite points of the plane α. To determine this distance we must obtain the line perpendicular to the plane a from the point P and get your point I intersection. Distance d of P to I It will be the minimum distance from this point to the plane a.
To solve this problem we can break down each of the necessary steps into elementary problems.:
- Determination of the notable lines (horizontal and frontal) of the plane.
- Determination of the normal direction to the plane a
- Obtaining the intersection point I of the perpendicular line passing through P with the plane a
- Determination of the true magnitude of the segment P-I that will be the distance sought.
Determination of notable lines
To simplify plots and get a clearer picture of the process, we will assume that the plane is defined by two of its notable straight lines: a horizontal “h” and a front “f” parallel to the Horizontal and Vertical projection planes respectively. The reader can see how to determine these lines from another definition of the plane in “Diédrico System: Straight lines in a plane parallel to the projection”
The problem statement With these data it will be:
Determine the minimum distance from the point P to a plane a defined by a horizontal and a front passing through the point A.
Determination of the line perpendicular to the plane
As we saw when studying the “Perpendicular to a plane“, the directions of the projections of the normal to a plane are perpendicular in each projection, to the lines of the plane parallel to said projection. In our case they will be perpendicular to the horizontal in the horizontal projection and to the front in the vertical. We will use the passing “P” and has the addresses listed.
Getting the point of intersection
The general method of obtaining intersections between lines and planes that have studied in the chapter “Incidence” It is based on using an auxiliary plane containing the straight. This plane will have a line of intersection “i” with the plane with which we get the intersection of the line. Point “I” It will be sought at the intersection “i” of the two planes
The chosen auxiliary plane may coincide with the plane normal projecting in any view. Hemos elegido la proyección vertical para determinar este plano auxiliar, ω, que produce una recta i de intersección con el plano formado con la horizontal y la frontal.
La recta de intersección “i” cortará a la frontal y a la horizontal en dos puntos (1 and 2).
La proyección que nos falta es inmediata de obtener al referir los puntos de intersección con las rectas notables en la proyección vertical a la proyección horizontal. The line “i” determinará el punto de corte “I” entre la recta y el plano.
Determinación de la verdadera magnitud de la mínima distancia
La forma de obtener la verdadera magnitud de un segmento la vimos en “Diédrico System: True magnitude of the line“. Necesitaremos construir un triángulo rectángulo cuyos catetos serán la proyección del segmento sobre un plano de proyección y la cota relativa de sus extremos medida en dirección perpendicular a este plano de proyección. La hipotenusa del triángulo nos determinará la verdadera magnitud del segmento.
En el caso de estudio que estamos desarrollando podemos determinar la cota relativa entre el punto “P” and “I”. Para ello obtendremos la cota relativa, z, de estos puntos respecto del plano horizontal. Esta magnitud la obtendremos en la proyección vertical según la dirección de las líneas de referencia entre las proyecciones vertical y horizontal.
Obtendremos la true magnitude mediante la construcción del triángulo rectángulo, obteniendo en consecuencia la distance “d” del punto P al plano α.
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