Visualising Solid Shapes

Drawing Solid Shapes on a Flat Surface (Sketches - Oblique and Isometric)

Representing three-dimensional objects on a two-dimensional plane, like a piece of paper or a screen, presents a challenge because we need to convey the perception of depth. Various drawing techniques have been developed for this purpose, and two common methods for sketching solid shapes are oblique sketches and isometric sketches. These techniques are particularly useful for visualising the basic structure of solids.

1. Oblique Sketches

An oblique sketch is a simple method for drawing 3D objects where one face of the object (usually the front face) is drawn true to its actual shape and size, as if you are looking directly at it. The depth of the object is then shown by lines drawn parallel to each other and extending back from the vertices of the front face at an angle.

• Front Face: The face parallel to the viewing plane (the paper) is drawn without any distortion, showing its true angles and side lengths. For a cuboid, the front face would be a rectangle with accurate dimensions.
• Receding Edges: The edges that represent the depth of the object are drawn parallel to each other and typically at a standard angle, often $45^\circ$, to the horizontal.
• Depth Lengths: A key characteristic of oblique sketches is that the lengths of these receding edges (showing depth) are usually not drawn to their true scale. They are often drawn shorter than the actual length to make the sketch look more natural and less elongated. If the depth is drawn at the full, true length, it's called a Cavalier projection; if it's shortened (e.g., to half length), it's a Cabinet projection. In general oblique sketches, these lengths are arbitrary or visually estimated.

Oblique sketches are relatively quick and easy to create as they do not require special paper or complex calculations. However, because the depth is often foreshortened and the side faces are skewed, they can sometimes appear distorted, especially for objects where depth is a significant dimension.

In the oblique sketch of the cube, the front face is a perfect square. The lines representing depth are parallel and drawn at an angle, but their drawn length is visually adjusted.

2. Isometric Sketches

An isometric sketch is another method for representing 3D objects, aiming to show the object's proportions more accurately from a specific perspective. 'Isometric' means 'equal measure', referring to the fact that lengths are drawn to scale along the three principal axes.

• Viewpoint: The object is viewed from an angle such that the three axes (representing length, width, and height) appear to be equally foreshortened and the angles between them are $120^\circ$.
• Axes Angles: In an isometric drawing on a flat surface, vertical lines of the object are drawn vertically. The lines representing the width and length of the object recede at an angle of $30^\circ$ from the horizontal baseline, one to the left and one to the right.
• Lengths: A significant advantage of isometric sketches is that all lines that are parallel to these three principal axes are drawn to their true relative lengths (i.e., to scale). If an edge is 5 cm long in the actual object and is parallel to one of the isometric axes, it is drawn as 5 units long in the sketch (where the unit is the chosen scale).
• Paper: Isometric sketches are often drawn using isometric dot paper or isometric grid paper, where the dots form a grid of equilateral triangles. This paper makes it easy to draw lines at the correct $60^\circ$ and $120^\circ$ angles (which appear as $30^\circ$ relative to the horizontal).

Isometric sketches provide a more visually balanced and proportionally accurate representation of the object compared to oblique sketches. While the angles in the sketch might not correspond to the true angles in the object (e.g., a right angle in the object might appear as $120^\circ$ or $60^\circ$ in the sketch), the lengths along the axes are maintained proportionally.

In the isometric sketch of the cube, the vertical edges are upright. The receding edges representing width and length are at $30^\circ$ to the horizontal, and crucially, all visible edges of the cube are drawn at the same length relative to each other, preserving proportion.

Comparison of Oblique and Isometric Sketches

Here is a summary comparing the key features of oblique and isometric sketching:

Both techniques are valuable for visualising 3D shapes, with the choice often depending on the desired emphasis (true front shape vs. overall proportions) and the ease of drawing.

Visualising Different Sections of a Solid (Cross-sections)

To gain a deeper understanding of the internal structure and properties of three-dimensional solid shapes, it is often helpful to imagine slicing through them. The two-dimensional shape that is revealed by such a slice is called a cross-section.

A cross-section is essentially the intersection of a 3D solid with a plane. If you think of slicing a solid object cleanly with an infinitely thin, flat plane (like cutting a loaf of bread), the shape formed on the cut surface is the cross-section.

The shape of the resulting cross-section is not fixed for a given solid; it depends critically on two factors:

• The intrinsic shape of the solid object itself.
• The angle and location of the slicing plane relative to the solid (i.e., where and how the cut is made).

We commonly consider cuts made in specific orientations:

• Vertical Cut: A cut made by a plane that is perpendicular to the base (or the resting surface) of the solid. Imagine slicing downwards through an object standing on a table.
• Horizontal Cut: A cut made by a plane that is parallel to the base (or the resting surface) of the solid. Imagine slicing across an object parallel to the table it is on.

Examples of Cross-sections from Common Solids

Let's explore the different cross-sections that can be obtained from various common solid shapes by making different cuts:

1. Cube/Cuboid:

   These are polyhedra with flat faces.

   • If you make a horizontal cut parallel to the base of a cube or cuboid, the cross-section will be a polygon identical in shape and size to the base face. For a cube, it's a square; for a cuboid, it's a rectangle.
   • If you make a vertical cut perpendicular to the base and parallel to a side face, the cross-section will be a polygon identical in shape and size to that side face. For a cube, it's a square; for a cuboid, it's a rectangle.
   • If you make a diagonal cut, the shape of the cross-section can vary. A cut through opposite vertices can yield a rectangle. A cut passing through more than four faces can even result in a hexagon.
   

2. Cylinder:

   This solid has curved surfaces and flat bases.

   • A horizontal cut made by a plane parallel to the circular bases will always result in a circle. If the plane passes through the center of the base, the cross-section circle will be the same size as the base circle.
   • A vertical cut made by a plane passing through the axis of the right circular cylinder and perpendicular to the bases will result in a rectangle.
   • A diagonal cut made by a plane that is not parallel to the bases and does not pass through the axis can result in an ellipse.
   • Specific cuts can also yield parabolas or pairs of parallel lines (when the cutting plane is parallel to the axis and tangent to the cylinder).
   

3. Cone:

   This solid has a curved surface, a flat base, and an apex.

   • A horizontal cut made by a plane parallel to the circular base will always result in a smaller circle (unless the cut is at the apex, which results in a point).
   • A vertical cut made by a plane passing through the apex and the centre of the base will result in a triangle (specifically, an isosceles triangle for a right circular cone).
   • Cuts made at other angles relative to the axis can produce shapes known as conic sections:
     • A cut parallel to a slant edge results in a parabola.
     • A diagonal cut that does not pass through the base results in an ellipse.
     • A vertical cut not through the apex results in a hyperbola.
   

4. Sphere:

   This is a solid with a single curved surface.

   • Any plane slicing through a sphere will always produce a circle as the cross-section.
   • If the cutting plane passes through the centre of the sphere, the resulting cross-section is the largest possible circle, called a great circle.
   

5. Pyramid (e.g., Square Pyramid):

   This solid has a polygonal base and triangular faces meeting at an apex.

   • A horizontal cut made by a plane parallel to the polygonal base will result in a smaller polygon that is similar in shape to the base. For a square pyramid, this results in a smaller square.
   • A vertical cut made by a plane passing through the apex and through the centre of the base will result in a triangle.
   • A vertical cut made by a plane parallel to one side of the base but not passing through the apex will result in a trapezium.
   

Visualizing and drawing cross-sections is a valuable skill for understanding the shape and structure of three-dimensional objects. It helps us connect the 3D form to the 2D shapes we are more familiar with.

Answer:

When a right circular cone is sliced by a plane parallel to its circular base, the resulting cross-section is always a shape similar to the base. Since the base is a circle, the cross-section will also be a circle. As the slicing plane moves further from the base towards the apex, the radius of the resulting circular cross-section decreases.

The shape of the cross-section is a circle.

Views of 3-D Shapes (Front View, Side View, Top View)

Another method for representing three-dimensional objects on a two-dimensional surface is by drawing their appearances from specific, standardized viewpoints. These representations, often referred to as orthographic projections, show the object as if viewed from an infinite distance along parallel lines perpendicular to the projection plane. The most commonly used orthographic views are based on looking directly at the front, side, and top of the object.

The three standard principal views are:

1. Front View: This view shows what the object looks like when observed from a direction perpendicular to its designated front face. It typically reveals the overall height and width of the object.
2. Side View: This view shows what the object looks like when observed from a direction perpendicular to one of its side faces. This is usually specified as either the Left Side View or the Right Side View. It typically reveals the overall height and depth (or width, depending on perspective) of the object.
3. Top View: This view, also known as the plan view, shows what the object looks like when observed from a direction directly above it. It typically reveals the overall width and depth of the object.

These views are drawn without perspective, meaning parallel lines in the object appear as parallel lines in the view, and lengths parallel to the viewing plane are shown true size. By presenting multiple views, engineers, architects, and designers can accurately communicate the complete shape and dimensions of a 3D object on a flat drawing.

Example 1. Consider a simple solid shape made of three identical cubes stacked in an 'L' shape. Two cubes are placed side-by-side on a base, and the third cube is placed on top of the rightmost of these two base cubes. Draw its front view, side view (from the right), and top view.

Answer:

Let's orient the shape such that the side with two cubes forming the base is considered the front. One cube is on the left in the bottom row, another is on the right in the bottom row, and the third is on top of the bottom right cube.

Front View:

When looking at the shape directly from the front, we see two square faces side-by-side at the bottom level. Above the right bottom square, we see the square face of the third cube.

The front view will consist of three squares arranged like the letter 'L'.

Side View (from the Right):

When looking at the shape directly from the right side, we see the rightmost cube in the bottom row. Directly on top of this cube, we see the third cube. The leftmost cube in the bottom row is hidden from this view.

The right side view will consist of two squares stacked vertically.

Top View:

When looking down at the shape directly from above, we see the top face of the leftmost cube in the bottom row. We also see the top face of the cube stacked above the rightmost bottom cube. These two top faces are side-by-side in the top view.

The top view will consist of two squares placed adjacent to each other.

By examining these different 2D views together, one can fully understand and mentally reconstruct the three-dimensional form of the object. This technique of using multiple views is a fundamental method in technical communication and design to represent 3D geometry accurately.