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Two-Point Perspective

Two-Point Perspective

Two-Point Perspective

Two-Point Perspective

Three-Point Perspective

Three-Point Perspective

Three-Point Perspective

Three-Point Perspective

Three-Point Perspective

The terms “1-point perspective,” “2-point perspective,” and“3-point perspective” refer to different types of pictures wecan encounter.The only difference lies in how the (usually rectangular)objects are oriented.(Demonstration with Rubik’s cube)

Where is the Viewpoint?

Every picture perspective has a perfect spotfrom which to view it.The trick is to find it, using clues in the picture.(Recall how we did it in 1-point perspective)

V1 V2

V1 V2

90◦

The Top View

x

E

The Top View

x

E

V1

The Top View

x

E

V1

V2

The Top View

x

E

V1

V2

The Top View

x

E

V1

V2

What We Need

x

V2

V1

What We Need

x

V2

V1

E

What We Need

What We Need

TheoremIf a picture is in two-point perspective, call the vanishing pointsV1 and V2. Then the correct viewing position is somewhere onthe semicircle in front of the picture connecting V1 and V2.

Two-Point Perspective 71

E

V1 V2

top ofbuildingparallelparallel

E

V1 V2

top of

building

parallelparallel

the eye-levelplane H

top edge ofpicture plane Figure 5.3. Two of many possible lo-

cations for the viewpoint E. Becausethe edges of the building form a rightangle, the lines of sight to the vanish-ing points must form a right angle atthe point E.

This brings up a question:

What is the set of all points E in the eye-level half-plane H such that EV1 and EV2 are perpendicular?

It turns out that this set is a semicircle whose endpoints are V1and V2 (see Figure 5.4).

V1

V2E

V1 V2

H

top view

Figure 5.4. The viewpoint E must lieon a horizontal semicircle in the half-plane H.

Theorem 5.1. The viewpoint E for a standard two-point per-spective painting (drawing, photograph) with vanishing points V1and V2 lies on a semicircle with endpoints V1 and V2. The planeof the semicircle is perpendicular to the picture plane.

Proof. Consider a possible viewpoint E in the half-plane H, as onthe left of Figure 5.5. Since E is a possible viewpoint, the lines EV1and EV2 are perpendicular. Let M be the midpoint of V1 and V2, sothat the two segments MV1 and MV2 have the same length r. Let

Consequences

ConsequenceThe further apart the two vanishing points, the larger theviewing circle, so the viewer can stand farther from the picture.

Another ConsequenceThis is one reason why great paintings always look better inperson than shrunk onto a computer screen or into the pagesof a textbook. When the image is shrunk, the viewing circleshrinks too!

Consequences

ConsequenceThe further apart the two vanishing points, the larger theviewing circle, so the viewer can stand farther from the picture.

Another ConsequenceThis is one reason why great paintings always look better inperson than shrunk onto a computer screen or into the pagesof a textbook. When the image is shrunk, the viewing circleshrinks too!

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2W1 W2

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2W1 W2

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

Finding the ViewpointSuppose this rectangle is actually a square.Where is the correct viewpoint?

V1 V2W1 W2

U

T

Answer: Stand directly in front of T ,at a distance equal to the distance between T and U.

How to Build a Cube in 2-Point Perspective

How to build a cube:1 Draw vertical lines through the corners of the square. The

four top corners will be somewhere on these lines.2 Measure the distance between V1 and U.3 Measure that same distance straight up from V1, and mark

that as a new vanishing point V ′.4 Now draw a diagonal from the near corner of the square up

to V ′. This is a diagonal of one side of the cube.5 Where this diagonal intersects one of the vertical sides of

the cube will be a corner.6 Use V1 and V2 to finish drawing the top square.

How to Build a Cube in 2-Point Perspective

How to build a cube:1 Draw vertical lines through the corners of the square. The

four top corners will be somewhere on these lines.2 Measure the distance between V1 and U.3 Measure that same distance straight up from V1, and mark

that as a new vanishing point V ′.4 Now draw a diagonal from the near corner of the square up

to V ′. This is a diagonal of one side of the cube.5 Where this diagonal intersects one of the vertical sides of

the cube will be a corner.6 Use V1 and V2 to finish drawing the top square.

How to Build a Cube in 2-Point Perspective

How to build a cube:1 Draw vertical lines through the corners of the square. The

four top corners will be somewhere on these lines.2 Measure the distance between V1 and U.3 Measure that same distance straight up from V1, and mark

that as a new vanishing point V ′.4 Now draw a diagonal from the near corner of the square up

to V ′. This is a diagonal of one side of the cube.5 Where this diagonal intersects one of the vertical sides of

the cube will be a corner.6 Use V1 and V2 to finish drawing the top square.

How to Build a Cube in 2-Point Perspective

How to build a cube:1 Draw vertical lines through the corners of the square. The

four top corners will be somewhere on these lines.2 Measure the distance between V1 and U.3 Measure that same distance straight up from V1, and mark

that as a new vanishing point V ′.4 Now draw a diagonal from the near corner of the square up

to V ′. This is a diagonal of one side of the cube.5 Where this diagonal intersects one of the vertical sides of

the cube will be a corner.6 Use V1 and V2 to finish drawing the top square.

How to Build a Cube in 2-Point Perspective

How to build a cube:1 Draw vertical lines through the corners of the square. The

four top corners will be somewhere on these lines.2 Measure the distance between V1 and U.3 Measure that same distance straight up from V1, and mark

that as a new vanishing point V ′.4 Now draw a diagonal from the near corner of the square up

to V ′. This is a diagonal of one side of the cube.5 Where this diagonal intersects one of the vertical sides of

the cube will be a corner.6 Use V1 and V2 to finish drawing the top square.

How to Build a Cube in 2-Point Perspective

How to build a cube:1 Draw vertical lines through the corners of the square. The

four top corners will be somewhere on these lines.2 Measure the distance between V1 and U.3 Measure that same distance straight up from V1, and mark

that as a new vanishing point V ′.4 Now draw a diagonal from the near corner of the square up

to V ′. This is a diagonal of one side of the cube.5 Where this diagonal intersects one of the vertical sides of

the cube will be a corner.6 Use V1 and V2 to finish drawing the top square.