Projections on Auxiliary Planes

Projections on Auxiliary Planes 
Sometimes none of the three principal orthographic views of an object show the different edges and faces of an object in their true sizes, since these edges and faces, are not parallel to any one of the three principal planes of projection.  In order to show such edges and faces in their true sizes, it becomes necessary to set up additional planes of projection other than the three principal planes of projection in the positions which will show them in true sizes.  If an edge or a face is to be shown in true size, it should be parallel to the plane of projection.  Hence the additional planes are set up so as to be parallel to the edges and faces which should be shown in true sizes.  These additional planes of projection which are set up to obtain the true sizes are called Auxiliary Planes. The views projected on these auxiliary planes are called Auxiliary Views.

The auxiliary view method may be applied

·        To find the true length of a line.

·        To project a line which is inclined to both HP and VP as a point.

·        To project a plane surface or a lamina as a line.

Types of auxiliary planes 
Usually the auxiliary planes are set up such that they are parallel to the edge or face which is to be shown in true size and perpendicular to any one of the three principal planes of projection.  Therefore, the selection of the auxiliary plane as to which of the principal planes of projection it should be perpendicular, obviously depends on the shape of the object whose edge or face that is to be shown in true size.

·        If the auxiliary plane selected is perpendicular to HP and inclined to VP, the view of the object projected on the auxiliary plane is called auxiliary front view and the auxiliary plane is called auxiliary vertical plane and denoted as AVP.

·        If the auxiliary plane is perpendicular to VP and inclined to HP, the view of the object projected on the auxiliary plane is called auxiliary top view and the auxiliary plane is called auxiliary inclined plane and denoted as AIP.

·        Auxiliary Vertical Plane (AVP) 
An AVP is placed in the first quadrant with its surface perpendicular to HP and inclined at Φ to VP. The object is assumed to be placed in the space in between HP, VP and AVP. The AVP intersects HP along the X₁Y₁ line.  The direction of sight to project the auxiliary front view will be normal to AVP.  The position of the auxiliary vertical plane w.r.t  HP and VP is shown in figure 1.

·        After obtaining the top view, front view and auxiliary front view on HP, VP and AVP, the HP, with the AVP being held perpendicular to it, is rotated so as to be in-plane with that of VP, and then the AVP is rotated about the X₁Y₁ line so as to be in plane with that of already rotated HP.

 Figure 1. The position of the auxiliary vertical plane w.r.t  HP and VP

·  
      Auxiliary Inclined Plane (AIP)

·        AIP is placed in the first quadrant with its surface perpendicular to VP and inclined at q to HP. The object is to be placed in the space between HP, VP and AIP.  The AIP intersects the VP along the X1Y1 line. The direction of sight to project the auxiliary top view will be normal to the AIP. The position of the AIP w.r.t  HP and VP is shown in figure 2.
After obtaining the top view, front view and auxiliary top view on HP, VP and AIP, HP is rotated about the XY line independently (detaching the AIP from HP). The AIP is then rotated about X1Y1 line independently so as to be in-plane with that of VP.

    Figure 2. The position of the AIP  w.r.t  HP and VP

·        Projection of Points on Auxiliary Planes

·        Projection on Auxiliary Vertical Plane

·        Point P is situated in the first quadrant at a height m above HP. An auxiliary vertical plane AVP is set up perpendicular to HP and inclined at Φ to VP. The point P is projected on VP, HP and AVP. 
As shown in figure 3, p' is the projection on VP, p is the projection on HP and P₁' is the  projection on AVP. 
Since point is at a height m above HP, both p' and p₁’ are at a height m above the XY and X₁Y₁ lines, respectively.

Figure 3. Projection of Point P on VP, HP and AVP

·        HP is rotated by 90 degree to bring it in plane of VP (figure 4(a) . Subsequently, the AVP is rotated about the X₁Y₁  line (figure 4(b),  such that it becomes in-plane with that of both HP and VP.

        Figure 4. The rotation of (a) HP and (b) AVP to make HP and AVP in plane with VP.

·        The orthographic projections (projections of point P on HP, VP and AVP) of point  P can be obtained be the following steps.

·        Draw the XY line and mark p and p', the top and front views of the point P.  Since AVP is inclined at Φto VP, draw the X₁Y₁ line inclined at  Φ to the XY line at any convenient distance from p.  Since point P is at a height m above HP, the auxiliary front view p₁' will also be at a height m above the X₁Y₁line.   Therefore, mark P₁’ by measuring o₁p₁’=op’ = m on the projector drawn from p perpendicular to the X₁Y₁ line.

Figure 5. Orthographic projection of the point P by Auxiliary projection method.

 Projection on Auxiliary Inclined Plane

·        Point P is situated in first quadrant at a distance n from VP.  An auxiliary plane AIP is set up perpendicular to VP and inclined at θ to HP. The point P is projected on VP, HP and AIP.

·        p' is the projection on VP, p is the projection on HP and P₁ is the projection on AIP. 
Since the point is at a distance n from VP, both p and p₁ are at a distance n above the XY and X₁Y₁lines, respectively

Figure 6. Orthographic projection of point P by auxiliary projection on AIP.

·        HP is now rotated by 90°  about XY line to bring it in plane with VP, as shown in figure 7(a). After the HP lies in-plane with VP, the AIP is rotated about the X₁Y₁, line, so that it becomes in-plane with that of both HP and VP. 
p and p’ lie on a vertical projector perpendicular to the XY line, and p’ and p₁ lie on a projector perpendicular to the X₁Y₁ line which itself is inclined at  θ  to XY line.

Figure 7. Orthographic projection of point P by auxiliary projection on AIP

The orthographic projections (projections of point P on HP, VP and AIP) of point  P can be obtained be the following steps.

·        Draw the XY line and mark p and p', the top and front views of the point P. 

·        Since AIP is inclined at q to HP, draw the X₁Y₁ line inclined at q to the XY line at any convenient distance from p’.

·        Since point P is at a distance n infront of VP, the auxiliary top view p₁ will also be at a distance n from the X₁Y₁ line. 

·        Therefore, mark P₁ by measuring o₁p₁=op = n on the projector drawn from p‘ perpendicular to the X₁Y₁ line.

Step by step procedure to draw auxiliary views

Auxiliary front view	Auxiliary top view
Draw the top and front views.	Draw the top and front views
Draw X1Y1   line inclined at  f (the inclination of AVP with VP) to the XY line.	Draw X1Y1 line inclined at  q (the inclination ofAIP with HP) to XY line.
Draw the projectors through the top views of the points perpendicular to the X1Y1 line.	Draw the projectors through the front views of the points perpendicular to the X1Y1 line.
The auxiliary front view of a point is obtained by stepping off a distance from the X1Y1 line equal to the distance of the front view of the given point from the XY line.	The auxiliary top view of a point is obtained by stepping off a distance from X1Y1  line equal to  the distance of the top view of the given point from the XY line

Auxiliary front view

Auxiliary top view

Draw the top and front views.

Draw the top and front views

Draw X₁Y₁   line inclined at  f (the inclination of AVP with VP) to the XY line.

Draw X₁Y₁ line inclined at  q (the inclination ofAIP with HP) to XY line.

Draw the projectors through the top views of the points perpendicular to the X₁Y₁ line.

Draw the projectors through the front views of the points perpendicular to the X₁Y₁ line.

The auxiliary front view of a point is obtained by stepping off a distance from the X₁Y₁ line equal to the distance of the front view of the given point from the XY line.

The auxiliary top view of a point is obtained by stepping off a distance from X₁Y₁  line equal to  the distance of the top view of the given point from the XY line

Auxiliary front view

Auxiliary top view

Draw the top and front views.

Draw the top and front views

Draw X₁Y₁   line inclined at  f (the inclination of AVP with VP) to the XY line.

Draw X₁Y₁ line inclined at  q (the inclination ofAIP with HP) to XY line.

Draw the projectors through the top views of the points perpendicular to the X₁Y₁ line.

Draw the projectors through the front views of the points perpendicular to the X₁Y₁ line.

The auxiliary front view of a point is obtained by stepping off a distance from the X₁Y₁ line equal to the distance of the front view of the given point from the XY line.

The auxiliary top view of a point is obtained by stepping off a distance from X₁Y₁  line equal to  the distance of the top view of the given point from the XY line