Plane APA’ is any plane in space with point (m, m’). The two red strings, the black string and the horizontal wire are all lines on the plane. Vertical and horizontal projections are given, as well as the results of rotation of the plane about lines AP, nm, cn, and ce.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Plane APA’ is given. The line perpendicular to the plane is represented by the oblique wire passing through (m, m’). The point of intersection of the line and the plane would be about halfway along the wire between (m, m’) and (c,0).
For more details, see COLL.1986.0885 and 1986.0885.01.01.
This relief shows the special case of a line that passes through the origin (0, 0, 0) and another point (a, b, c). The line is represented by the wire.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Plane APA’ is a plane perpendicular to the horizontal plane with point (m, m’) on APA’ (at corner of wire). Vertical and horizontal projections are shown, as well as rotation about line AP in the horizontal plane.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
The three points in space are represented at the bends in the three wires at points (a, a’), (b, b’) and (c, c’). The red lines connect the points in pairs showing the resulting triangle that lies on the plane that was to be constructed.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Planes APA’ and BQB’ (where A=B=c and A’=B’=d’) are given. Then line d’c’ is the vertical projection of the intersection of the two planes while line dc is the horizontal projection of the intersection.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
The plane APA’ is intersected by line bc’ represented by the black string. The red string represents a line on the plane which bc’ intersects at point (m, m’). Horizontal and vertical projections of these lines are shown.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Planes APA’ and BQB’ are parallel in the horizontal plane (see their respective horizontal projections AP and BQ) and intersect along line (o, c’)-(d, d’) (wire). This intersection is also parallel to the horizontal projections of the two planes (observe that cd is also parallel in the horizontal plane).
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Figure 1 shows the rotation and projections of point (m, m’) about a horizontal line perpendicular to the vertical plane through point a’. Figure 2 shows the rotation and projections of point (m, m’) about the vertical line perpendicular to the horizontal at point a.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Two projections are shown. The left shows the vertical and horizontal projection of a line. The right shows a line rotated to the horizontal and vertical plane at a point of contact with each plane to show the angle of the line with each plane.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Three points, a, b, and c on the horizontal plane make the triangular base of the pyramid. Point (s, s’) at the bend in the wire will be the apex. The three black strings represent the three remaining edges of the pyramid. Notice that the apex is not above the base. The various projections after rotation to the horizontal plane allow the lengths of the sides to be found. The height of the pyramid is segment sS_{3}.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Figure 1 on the left shows plane APP’ cutting obliquely through the space with the black string showing a line on the plane. The vertical and horizontal projections of the plane are also depicted.
Figure 2 on the right shows the vertical and horizontal projection of a line in the plane BQB’.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
As with relief 17, APA’ is again the plane and the red string de’ is on the plane. In this relief, the line is represented by the wire coming out of the horizontal plane and away from the vertical plane (it intersects the vertical plane below the horizontal plane). The point of intersection is at (m, m’) where the wire, the string and the bent wire meet. The horizontal and vertical projections are shown.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Plane APA’ (right construction) and line through (m, m’) parallel to the plane and to line cd’ on the plane are given (red strings). Then the plane BQB’ parallel to plane APA’ is constructed.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
The point (m, m’) on the left side of the relief is given. On the left side, two lines are given: ab’ depicted by the black string, and dc’ (black string missing). By constructing the red lines hg’ and ef’ parallel to lines dc’ and ab’ respectively, the plane PQP’ containing the point (m, m’) is formed parallel to the two given lines.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Planes CDD’C’ and ABB’A’ are both parallel to the x-axis (crease in the card). They intersect in line (e, e’)-(f, f’) (wire) which is also parallel to the x-axis. The planes can be visualize by imagining both red strings extending left and right. Both projections of this intersection are shown as well as the rotation of it about the horizontal line perpendicular to the x-axis PA.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
Point (m, m’) is rotated about line ab. Point n on the horizontal plane is the foot of the perpendicular from the point to line ab. Point M_{1} is the result of rotation of the point about line nm; M_{2} is the result of rotation about line ab.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
The black string represents the given line bc’, while the point (a, a’) at the bend of the wire represents the given point. The horizontal line coming out of the vertical plane denoted cd’ is perpendicular to line bc’. Point P is the intersection of the vertical and horizontal projections of the wire with the x-axis. It follows that plane FPF’ which contains the line (c,0)-(0,d’) (wire) is also perpendicular to line bc’.
For more details, see COLL.1986.0885 and 1986.0885.01.01.
The horizontal line is represented by the wire coming out of the vertical plane at c’. The second line would run from point a on the horizontal plane to b’ on the vertical, however the black string that should represent this line is missing. The two lines intersect at point (m, m’) at the bend in the wire. Lines cd and ab are the projections of the lines on the horizontal plane, with line nm perpendicular to cd at m. By rotating about these three lines as in relief 25, the angle between the given lines is shown on the horizontal plane as angle aM_{2}D_{1}.
For more details, see COLL.1986.0885 and 1986.0885.01.01.