PROJECTIVE GEOMETRY

Projective Geometry

• enables a clearer understanding of some more generic properties of geometric objects.
• is a non-Euclidean geometry that formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way.

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History

• early Italian Renaissance
• architectural drawings:
• Filippo Brunelleschi (1377–1446)‏
• Leon Battista Alberti (1404–1472)‏

• invented the method of perspective drawing.

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History

Definition of Terms:

• Plane – the imaginary flat surface with length and width but no thickness, it may extend indefinitely in any direction.
• Station point (SP) – the point where the observer is supposed to stand while viewing the object.
• Picture plane (PP) – the plane in which the object is located.
• Horizon – the horizontal plane at level.

• Vanishing point (VP) – a point wherein the edges of the object will imaginarily meet.
• Vanishing line (VL) – the rays from the vanishing point from the edges which serves as the guide in locating the ends of the object.
• Ground plane (GP) – a plane which is in-line with the eye.
• Ground line (GL) – the intersection of the ground plane and the picture plane.

Basic elements:

• Points
• Lines
• Planes

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Basic elements:

• Points

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Basic elements:

• Points

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Basic elements:

• Points

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• Lines

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Basic elements:

• Points
• Lines

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• Planes

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Geometric Transformations

• Reflections - is a type of transformation where the line of symmetry is a perpendicular bisector of corresponding points.
• Translations - preserve congruency. This means the image and pre-image of an object is exactly the same shape and size as the pre-image .
• Rotation - moves all the points of the figure the same angle around the same pivot point.
• Dilations - is a scaled transformation. The distance from each point in the pre-image to the center is multiplied by the scale factor to find the distance along in the same direction to the corresponding image point.

Reflections

Reflections

• The pre-image ABC is reflected onto the image A'B'C' across the x-axis.

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Reflections

Translations

Translations

• The pre-image is translated onto the image 4 units right.

Rotations

Rotations

• The pre-image ABC is rotated to image A'B'C' about P(0,0) as the center of rotation and through 90° as the angle of rotation.

Dilations

Dilations

• The triangle is transformed by a dilation with the center at (0,0) and a scale factor of 2.

Axonometric Projection

• the ability to show the inclined position of an object with respect to the plane of projection.

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• Kinds:
• Isometric – Iso (one or equal) and Metrus (measures); equal measures.
• Dimetric – an axonometric drawing into two angle.
• Trimetric – utilizes three different angles.

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Axonometric Projection

Isometric Projection

• In isometric projection the angles between the projection of the axes are equal i.e. 120º.

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• It is important to appreciate that it is the angles between the projection of the axes that are being discussed and not the true angles between the axes themselves which is always 90º.

Isometric Projection

Isometric Projection

Isometric Projection

Isometric Projection

• The angles between the projection of the axes in dimetric projection renders two of the three to be equal.
• To draw the outline of an object in dimetric projection, two scales are required.
• The scales are generated the same as for isometric.

Dimetric Projection

Dimetric Projection

Dimetric Projection

Trimetric Projection

• In trimetric projection the projection of the three angles between the axes are unequal.

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• Thus, three separate scales are needed to generate a trimetric projection of an object.

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• The scales are constructed using the same method described in isometric and dimetric projection.

Trimetric Projection

Trimetric Projection

Orthographic Projection

• Orthographic projection shows complex objects by doing a 2D drawing of each side to show the main features.
• Orthographic drawings usually consist of a front view, a side view and a top view, but more views may be shown for complex objects with lots of detail.

• Here are three orthographic views of an object.

Orthographic Projection

Orthographic Projection

Orthographic Projection

Perspective Projection

• from Latin perspicere which means “to see clearly”.
• is the most attractive type of presenting an object, the subject appears as it seen by the naked eye.
• It is a photographic or “picture like” result.

Perspective Projection

One Point Perspective

• Also called parallel perspective.
• Occurs when one of its faces is parallel to the plane of projection.
• Has only one vanishing point used.

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One Point Perspective

Even though change in eye position or tilt of head affects the vanishing point, the view is still one point perspective.

Even though change in eye position or tilt of head affects the vanishing point, the view is still one point perspective.

Two Point Perspective

• Also called angular perspective.
• If it employs two vanishing points and the sides are angular with the picture plane.

Two Point Perspective

Three Point Perspective

• In oblique perspective, three vanishing points are employed.
• If the projection plane is not parallel to any principal axis, a three-point projection occurs.
• Only the edges are perpendicular with the plane of projection and will show its true dimension.

Three Point Perspective

THE END