Auxiliary Projection

Introduction

In engineering drawing, orthographic projections are the standard method to represent three-dimensional objects on two-dimensional planes. Typically, we use three principal views: front, top, and side. These views are projected onto planes that are perpendicular to each other, providing clear and measurable representations of the object.

However, orthographic views have limitations. When a surface of an object is inclined to all principal planes, its true size and shape are not visible in the standard views. Instead, the inclined surface appears foreshortened or distorted. This makes it difficult to understand the exact geometry or to dimension the feature accurately.

To overcome this, auxiliary projection is used. Auxiliary views are additional views projected onto planes that are auxiliary (or supplementary) to the principal planes, specifically oriented to be perpendicular to the inclined surface. This allows the true size and shape of the inclined surface to be seen clearly.

Auxiliary projection is essential in engineering drawing to accurately represent complex features, such as inclined holes, chamfers, or sloping faces, which cannot be properly visualized in standard orthographic views.

Purpose and Principle of Auxiliary Projection

The principle of auxiliary projection is to project the object onto a plane that is not one of the principal planes (front, top, or side), but an auxiliary plane positioned perpendicular to the inclined surface of interest.

By doing this, the inclined surface is viewed "face-on," revealing its true size and shape without distortion. This is because the projection plane is oriented exactly perpendicular to the surface, eliminating foreshortening.

Let's clarify some terms:

Principal planes: The three mutually perpendicular planes used in orthographic projection - frontal (vertical), horizontal, and profile (side) planes.
Auxiliary plane: A plane placed at an angle to the principal planes, specifically perpendicular to the inclined surface whose true size is to be shown.
Primary auxiliary view: A view projected onto an auxiliary plane perpendicular to one of the principal planes.
Secondary auxiliary view: A view projected onto an auxiliary plane perpendicular to a primary auxiliary plane, used when surfaces are inclined to more than one plane.

Why is this necessary? Because an inclined surface, when projected onto a principal plane, appears shorter than its actual size due to the angle. Only by projecting it onto a plane perpendicular to it can we see its true dimensions.

Types of Auxiliary Views

Auxiliary views are classified based on the plane from which they are projected:

Primary Auxiliary View: This is projected perpendicular to one of the principal planes (front, top, or side). It is used when the inclined surface is directly related to a principal plane. For example, if a surface is inclined to the frontal plane, the primary auxiliary view is projected perpendicular to the frontal plane to show the true size of that surface.
Secondary Auxiliary View: Sometimes, surfaces are inclined to both a principal plane and an auxiliary plane. In such cases, a secondary auxiliary view is projected perpendicular to the primary auxiliary plane to reveal the true size of the surface. This is a further step to handle complex geometries.

Understanding these types helps in choosing the correct auxiliary plane and projection direction to accurately represent the object.

Steps to Construct Auxiliary Projection

Constructing an auxiliary view involves a systematic approach to ensure accuracy and clarity. The following steps outline the process:

graph TD    A[Identify the inclined surface] --> B[Draw the auxiliary plane perpendicular to the inclined surface]    B --> C[Project points from the principal view onto the auxiliary plane]    C --> D[Connect projected points to form the auxiliary view]    D --> E[Dimension the auxiliary view to show true size]

Step 1: Identify the Inclined Surface
Examine the orthographic views to find the surface that is inclined and whose true size is not visible.

Step 2: Draw the Auxiliary Plane
Construct an auxiliary plane perpendicular to the inclined surface. This plane is usually drawn at an angle to the principal planes.

Step 3: Project Points
From the known points on the inclined surface in the principal view, project lines perpendicular to the auxiliary plane. Mark the intersection points on the auxiliary plane.

Step 4: Complete the Auxiliary View
Join the projected points in sequence to form the true shape of the inclined surface on the auxiliary plane.

Step 5: Dimension the Auxiliary View
Add dimensions to the auxiliary view to communicate the true size and shape of the inclined surface clearly.

Worked Example 1: Auxiliary View of a Single Inclined Surface

Example 1: Auxiliary View of a Single Inclined Surface Easy

Construct the auxiliary view of a rectangular plate inclined at 30° to the horizontal plane. The plate measures 100 mm by 60 mm.

Step 1: From the top view, identify the inclined edge of the plate. Since the plate is inclined at 30° to the horizontal plane, the top view shows the foreshortened length.

Step 2: Draw the auxiliary plane perpendicular to the inclined surface. This plane will be inclined at 30° to the horizontal plane.

Step 3: Project points from the top view perpendicularly onto the auxiliary plane. For example, mark the corners of the plate and draw projection lines at 30° to the horizontal.

Step 4: Connect the projected points to form the auxiliary view. This view will show the true length (100 mm) and width (60 mm) of the plate without distortion.

Answer: The auxiliary view accurately represents the plate's true size, confirming the length as 100 mm and width as 60 mm.

Worked Example 2: Primary Auxiliary View of an Inclined Surface

Example 2: Primary Auxiliary View of an Inclined Surface Medium

Draw the primary auxiliary view of a prism with one face inclined at 45° to the frontal plane. The prism has a base of 80 mm and height of 50 mm.

Step 1: Identify the inclined face in the front view, which appears foreshortened.

Step 2: Draw the auxiliary plane perpendicular to the frontal plane and inclined at 45° to it.

Step 3: Project points from the front view onto the auxiliary plane using perpendicular projection lines.

Step 4: Connect the projected points to form the auxiliary view, showing the true shape and size of the inclined face.

Answer: The auxiliary view reveals the true dimensions of the inclined face, enabling accurate measurement and dimensioning.

Worked Example 3: Secondary Auxiliary View for Complex Inclined Surfaces

Example 3: Secondary Auxiliary View for Complex Inclined Surfaces Hard

A surface is inclined at 30° to the frontal plane and 45° to the horizontal plane. Construct the secondary auxiliary view to find the true shape of the surface.

Step 1: Draw the primary auxiliary plane perpendicular to the frontal plane and inclined at 30°.

Step 2: Project the inclined surface onto this primary auxiliary plane to get the primary auxiliary view.

Step 3: Draw the secondary auxiliary plane perpendicular to the primary auxiliary plane and inclined at 45°.

Step 4: Project points from the primary auxiliary view onto the secondary auxiliary plane to obtain the secondary auxiliary view.

Step 5: Connect the points to reveal the true shape of the surface.

Answer: The secondary auxiliary view shows the true size and shape of the surface inclined to both principal and auxiliary planes.

Worked Example 4: Application in Engineering Component

Example 4: Application of Auxiliary Projection in Engineering Drawing Medium

A mechanical part has an inclined hole drilled at 60° to the horizontal plane. Use auxiliary projection to show the true size of the hole and dimension it.

Step 1: Identify the inclined hole in the top and front views.

Step 2: Draw the auxiliary plane perpendicular to the axis of the hole, inclined at 60° to the horizontal plane.

Step 3: Project the hole's outline from the principal views onto the auxiliary plane.

Step 4: Draw the auxiliary view showing the true diameter and shape of the hole.

Step 5: Add dimensions for the hole diameter and position on the auxiliary view.

Answer: The auxiliary view accurately represents the inclined hole, facilitating correct manufacturing and inspection.

Worked Example 5: Dimensioning Using Auxiliary Projection

Example 5: Dimensioning Using Auxiliary Projection Easy

Demonstrate how to dimension an inclined surface of length 120 mm at 40° to the horizontal plane using auxiliary projection.

Step 1: Draw the inclined surface in the principal view, noting the foreshortened length.

Step 2: Construct the auxiliary plane perpendicular to the inclined surface at 40°.

Step 3: Project the endpoints of the inclined surface onto the auxiliary plane.

Step 4: Draw the auxiliary view showing the true length of 120 mm.

Step 5: Add dimension lines and labels on the auxiliary view to clearly indicate the true size.

Answer: The auxiliary view provides an accurate dimensioning reference for the inclined surface.

Formula Bank

True Length of an Inclined Line

\[ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

where: \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\) are coordinates of the line's endpoints in millimeters.

Tips & Tricks

Tip: Always draw the auxiliary plane perpendicular to the inclined surface for true size representation.

When to use: When constructing auxiliary views to avoid distorted projections.

Tip: Use light construction lines and extend projection lines carefully to maintain accuracy.

When to use: During step-by-step drawing to prevent clutter and errors.

Tip: Memorize the sequence: identify inclined surface -> draw auxiliary plane -> project points -> complete view.

When to use: To speed up drawing during timed competitive exams.

Tip: Check angles with a protractor or use CAD tools for practice to understand auxiliary planes better.

When to use: While learning or verifying manual drawings.

Tip: Compare auxiliary views with orthographic views to cross-verify dimensions and shapes.

When to use: For error checking and ensuring accuracy.

Common Mistakes to Avoid

❌ Drawing the auxiliary plane not perpendicular to the inclined surface.

✓ Ensure the auxiliary plane is exactly perpendicular to the inclined surface to get true size.

Why: Students often confuse the auxiliary plane orientation, leading to distorted views.

❌ Projecting points incorrectly by mixing projection directions.

✓ Maintain consistent projection direction perpendicular to the auxiliary plane.

Why: Inconsistent projection lines cause inaccurate auxiliary views.

❌ Not extending projection lines fully to intersect the auxiliary plane.

✓ Extend projection lines sufficiently to meet the auxiliary plane for accurate point location.

Why: Short projection lines result in incomplete or incorrect auxiliary views.

❌ Confusing primary and secondary auxiliary views.

✓ Remember primary auxiliary views are projected from principal planes; secondary are from auxiliary planes.

Why: Misunderstanding leads to incorrect construction sequences.

❌ Neglecting to dimension the auxiliary view properly.

✓ Always add dimensions on the auxiliary view to communicate true size clearly.

Why: Auxiliary views are often used for dimensioning; omission reduces drawing clarity.

Projection Type	Purpose	Construction	Applications
Orthographic	Show multiple views on principal planes	Project perpendicular to principal planes	Basic engineering drawings, manufacturing
Auxiliary	Show true size of inclined surfaces	Project onto planes perpendicular to inclined surfaces	Complex features, inclined holes, chamfers
Isometric	3D pictorial view with equal scale	Axes at 120°, no foreshortening	Visualisation, assembly drawings
Perspective	Realistic 3D representation	Vanishing points, converging lines	Presentations, design concepts
Oblique	Quick 3D representation with one face true size	Project front face true size, others at angle	Preliminary sketches, simple parts

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Auxiliary Projection