Intersection of a line and a vertical/horizontal/profile projection plane The horizontal projection of the intersection point N of the line p and the plane P lies on the 1st trace of the plane. Why? N’’ p’’ r2 x r1 N’ p’

Intersection of a line and a plane determined with its traces q’’ Q2’’ Outline of the solution: r2 p’’ 1. p   Becouse of the simplicity of the construction the plane  is a vertical projection plane. N’’ Q1’’ 1x2 2.    = q Q2’ = d1 3. q  p = N N’ Q1’ p’ = q’ r1

Exercises. 1. Determine the intersection of the line p and the plane . z p’’’ B’’’ B’ B’’ Remark. A line parallel with 3 is not uniquely determined by its horizontal and vertical projection. It has to be determined with the projections of two of its points. s3 p’= p’’ N’’’ N’’ A’ A’’ x A’’’ y N’ s1 y s2

2. Construct a line segment of the length d lying on the line p starting from the intersection with the plane P. d d2 d1  q” r2 p” S” Q1’ K’ K” T” q’ Q2” Q2’ Q1” x S’ r1 T’ p’ p0 S0 T0 K0

Two basic constructions (perpendicularity) 2. Construct a plane through a given point perpendicular to a given line. 1. Construct a line perpendicular to a given plane through a given point. x p’’ p’ T’’ T’ . s” r1 r2 r2 r1 x T’’ T’ n’ n’’ . S1’ S1” s’ Instruction. The 2nd principle line (or 1st princile line) is used to construct the traces of a plane passing through the point T.

Exercise 3. Construct the symmetry plane of the line segment AB. Definition. The set of point in space which have the same distance from the end-points of a line segment lie in the symmetry plane of that line segment. M1” M1’ s1 . Symmetry plane passes through the midpoint of the line segment and is perpendicular to the line segment.

Exercise 4. Construct a plane  perpendicular to the plane P and passes through line p. Remark! A plane is perpendicular to a plane is one of it containes at least one line perpendicular to the other plane.

Metrical exercise 1. Determine the distance between the point T and the plane P. d N0 T0 Outline of the solution: 1) T n, n  P N’’ r2 n’ n’’ 2) n  P = N q’’ N’ T’’ d1 d2 3) d (T,N) x r1  q’ T’

2. Construct line segment of lenght d from point S on a line perpendicular to the plane P. V’’ r2 s’’ S’’ x s’ S’ n0 . p0 S0 d V’ r1 V0

3. Determine the distance between point A and the plane . x y z A’’’ s2 d . s3 Remark 1. The plane  is the profile projecting plane. N’’ N’’’ N’ Remark 2. This construction is the same for the following task: Construct a line segment of given lenght from a point in a plane on a perpendicular line to the plane. A’ s1

Solved exercises p’ 1. Intersection of a trinagle and a line p’’  d2 Instruction. Because of the simplicity of the construction a vertical projection plane  through the line is used, and afterwards its intersection with the triangle plane is constructed. p’’  d2  q’’ M’’ N’’ P’’ B’’ A’’ x q’ d1 C’ M’ N’ B’ P’ Remark. The inetrsection of a parallelogram and a line is constructed in the same way. p’ A’

2. Intersection of the line p and a plane determined with the intersecting lines (a, b) q’’ b’’ N’’ 1) p    1 Pravcem je postavljena prva projicirajuća ravnina . 1’’ M’’ 2’’ 2) P   = q Pravci a i b probadaju ravninu  u točkama 1 i 2, a njihova je spojnica presječnica q ravnina P i . a’’ 3) q  p = N M’ a’ 2’ Napomena. Na isti se način konstruira probodište pravca i ravnine zadane dvama paralelnim pravcima. 1’ N’ p’ =d1 =q’ b’

3. Construct the intersection of the line p (parallel to the x axis) and the plane .  d2  q’’ S’’ The intersection of the vertical projection plane  through the line p and the plane  is the principal line q of the plane  . q’ S’

4. Intersection of a line and the symmetry plane/coincidence plane with the use of the profile peojection. z p’’’ P2’’’ P2” s3 p” N’’’ N” x P1’’’ P1” N’ s1  s2  k1  k2 P2’ p’ P1’ R’’’ R’= R” k3 y N = p   R = p  K

1. Determine the distance between the point T and the line p. Metrical exercises. 1. Determine the distance between the point T and the line p. x T’ T’’ p’’ p’ r2 d1 d2  Outline of the solution: 1) T  P, P  p 2) n  P = N 3) d (T,N) N’’ r1 d T0 N’

2. Construct a line segment of lenght d from the point A on a line perpendicular to the plane P. z r3 d B’’’ n’= n” B” n’’’ d A” A’’’ r2 r1 B’ x A’ Remark. There exist two solutions. y