Projections of Points and Lines Inclined to Both Planes

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1. Projections of Points Across the Four Quadrants

A point in space is denoted by capital letter $P$. Its projections are:

• Top View ($p$): Formed on the HP. The distance of $p$ from the $XY$ line equals the distance of point $P$ from the VP.
• Front View ($p'$): Formed on the VP. The distance of $p'$ from the $XY$ line equals the distance of point $P$ from the HP.

                2nd Quadrant             |           1st Quadrant
            Both p' and p ABOVE XY       |   p' ABOVE XY  |  p BELOW XY
      -----------------------------------+----------------------------------- XY
            Both p' and p BELOW XY       |   p' BELOW XY  |  p ABOVE XY
                4th Quadrant             |           3rd Quadrant

Quadrant	Spatial Location of Point $P$	Front View ($p'$) Position	Top View ($p$) Position
$1^{\text{st}}$	Above HP, In front of VP	Above $XY$ (by height above HP)	Below $XY$ (by distance in front of VP)
$2^{\text{nd}}$	Above HP, Behind VP	Above $XY$	Above $XY$
$3^{\text{rd}}$	Below HP, Behind VP	Below $XY$	Above $XY$
$4^{\text{th}}$	Below HP, In front of VP	Below $XY$	Below $XY$
On HP	In HP, distance $d$ in front of VP	On the $XY$ line	Below $XY$ by distance $d$
In VP	In VP, height $h$ above HP	Above $XY$ by height $h$	On the $XY$ line

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2. Straight Lines: Essential Terms & Notations

When a straight line $AB$ is inclined to both reference planes (HP and VP):

• True Length ($TL$ or $L$): The actual 3D length of line segment $AB$.
• True Inclinations:
  • $\boldsymbol{\theta}$ = True angle made by the line with the Horizontal Plane (HP).
  • $\boldsymbol{\phi}$ = True angle made by the line with the Vertical Plane (VP).
• Apparent Lengths:
  • $\boldsymbol{a'b'}$ or $\boldsymbol{EL}$ = Elevation Length (length of Front View on VP).
  • $\boldsymbol{ab}$ or $\boldsymbol{PL}$ = Plan Length (length of Top View on HP).
• Apparent Inclinations:
  • $\boldsymbol{\alpha}$ = Apparent angle made by the Front View with the $XY$ line.
  • $\boldsymbol{\beta}$ = Apparent angle made by the Top View with the $XY$ line.

Cardinal Relationships:

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3. The Master Projection Diagram (Method of Rotation)

        Locus of b' ------------------------b'-----------b1' (at angle theta, length TL)
                                           /|
                                          / |
                     a'                  /  |
        Locus of a' -+------------------+   |
                     |                 /    |
=====================|================/=====|====================== XY
                     |               /      |
        Locus of a  -+--------------+       |
                     a             /        |
                                  /         |
        Locus of b  -------------b----------b2 (at angle phi, length TL)

The 12 Fundamental Parameters of a Line Problem:

1. True Length ($TL$)
2. True inclination with HP ($\theta$)
3. True inclination with VP ($\phi$)
4. Front View Length / Elevation Length ($EL$)
5. Top View Length / Plan Length ($PL$)
6. Apparent inclination of Front View ($\alpha$)
7. Apparent inclination of Top View ($\beta$)
8. Distance of point $A$ above HP / below HP
9. Distance of point $A$ in front of VP / behind VP
10. Distance of point $B$ above HP / below HP
11. Distance of point $B$ in front of VP / behind VP
12. Distance between end projectors ($D$)

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4. Standard Construction Procedure (Rotation Method)

Given: End $A$ position ($h_a$ above HP, $d_a$ in front of VP), True Length $TL$, $\theta$, and $\phi$.

1. Draw Reference and End $A$:
   • Draw horizontal reference line $XY$.
   • Draw vertical projector for $A$: mark $a'$ at distance $h_a$ above $XY$, and mark $a$ at distance $d_a$ below $XY$.
2. True Inclination with HP ($\theta$):
   • Through $a'$, draw a line at angle $\theta$ to the horizontal of length equal to $TL$.
   • Label the endpoint $b_1'$.
   • Draw a horizontal line through $b_1'$: this is the Locus of $b'$.
   • Project $b_1'$ vertically down to the horizontal line passing through $a$ to get point $b_1$.
   • The length $a b_1$ is the Plan Length ($PL$).
3. True Inclination with VP ($\phi$):
   • Through $a$, draw a line at angle $\phi$ to the horizontal of length equal to $TL$.
   • Label the endpoint $b_2$.
   • Draw a horizontal line through $b_2$: this is the Locus of $b$.
   • Project $b_2$ vertically up to the horizontal line passing through $a'$ to get point $b_2'$.
   • The length $a' b_2'$ is the Elevation Length ($EL$).
4. Drawing Final Projections:
   • With center $a'$ and radius $a' b_2' = EL$, swing an arc to intersect the Locus of $b'$ at point $b'$.
   • Join $a'b'$ with a dark line. This is the Front View (angle with horizontal $= \alpha$).
   • With center $a$ and radius $a b_1 = PL$, swing an arc to intersect the Locus of $b$ at point $b$.
   • Join $ab$ with a dark line. This is the Top View (angle with horizontal $= \beta$).
5. Universal Sanity Check:
   • The endpoints $b'$ and $b$ must lie on the exact same vertical line (common end projector)!

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5. Summary of Key Exam Points

1. Elevation Length vs True Length: $EL < TL$, $PL < TL$.
2. Angles Inequality: $\alpha > \theta$ and $\beta > \phi$.
3. Collinear Projector Rule: The projectors joining $a'$ with $a$, and $b'$ with $b$, must always be strictly perpendicular to $XY$.
4. Parallelism Rule: If a line is parallel to HP, its front view is parallel to $XY$, and its top view shows the true length ($PL = TL$).
5. Sum of Angles ($\theta + \phi$): In any real oblique line, $\theta + \phi < 90^\circ$. If $\theta + \phi = 90^\circ$, the line lies in a profile plane.