Math 314

Spatial Sense

Topics

• Views• Drawing Methods

- Oblique- Opaque- Axiometric- Single Vanishing- Double Vanishing

• Solids

Views

• We are going from two dimensional to three dimensional.

• Lines have 1 dimension – length• Surfaces have two dimensions length

and height or length and width. • By combining surfaces we can construct

shapes that define a space called solids. They have three dimensions – length width & height.

2 dimensional

BACK

RIGHT

FRONT

LEFT

Adding Cubes

• Now we can place cubes on the grid. The numbers represent the number of cubes.

• For the front view put one cube on the left side (1st row, 1st column)

• Put 2 cubes on the left side (2nd row, 1st column).

Cube Viewing

3 1 2

2 1 2

1 1 1

BACK

RIGHT

FRONT

LEFT

Key for Success

• The key to these exercises is to ask yourself what is the most amount of cubes you see in a specific column

• Be careful when looking at rows and columns; it can be confusing!

Drawing a Front View

3 1 2

2 1 2

1 1 1

BACK

RIGHT

FRONT

LEFT

•As we look at the left-most column we see 3 cubes

•Middle-1 cube

•Right – 2 cubes

Front View

Draw a right view

• Again, ask yourself… ask yourself what is the most amount of cubes you see in a specific column

Recall Original Figure

3 1 2

2 1 2

1 1 1

BACK

RIGHT

FRONT

LEFT

Right View

• 1st column = 1• 2nd column = 2• 3rd column = 3

Homework

•#1 a-j (Stencil)

#1 a

2 3 1

3 2 1

1 1 1

Homework Solutions 1a

Front Right

Work in Class

• P17 a1); the answer is… • Left• A2); the answer is… • Top• A3; the answer is… • Front• A4; the answer is… • Bottom

Work in Class

• P18 c1 Front• 3,1,2 – draw on board• C2 right• 3,2• C 3 back• 2,1,3• C4 (top)• 2,elevated 1,elevated 1 (draw on

board!)

Work in Class / Homework

• P19 e 1-6• P20 g 1-6

Going backwards

• If we are given a front and right view can we give a top grid view?

• In most cases, the view is not unique (more than one answer)

• The trick is to start with 1’s (not 3’s) and move up…

Front Right

GIVE TOP GRID VIEW

Top View

3 1 1/2

1/2 1 1/2

1 1 1

BACK

RIGHT

FRONT

LEFT

Work in Class / Homework

• Stencil #2 a-e• Do one more on board… 2a• 1ST 5 people to come see me for

points!

#2a Solutions

3 1 / 2 1

1 1 1

3 1 / 2 1

Drawing a Cube

• We want to add a little more reality into our drawings

• Oblique • Face parallel to the paper’s edges• Receding lines reduced size (usually

half)• Face is in the foreground• Disappears to the right; RIGHT facing

Drawing a Cube

Definitions - Prism

• Between two parallel planes and a polyheatron

• A congruent (equal) polygon in each plane called bases

• Lateral shapes are rectangles

Definition - Pyramid

• Between two parallel planes and a polyhedron

• A polygon in one plane called a base

• An apex in the other plane• Lateral shapes are triagles

Cylinder

• Between two parallel planes and a curved surface

• A congruent circle in each parallel plane called base

• Continual lateral shape of a rectangle

Cone

• Between two parallel planes and a curved surface

• A circle in one place • An apex in the other• A lateral shape of a circle with an

angle cut out of it.