Todays programme

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Todays programme

• Important points and lines.
• Orthographic projection features.
• Orthographic projection onto two planes
• point, line, plane.
• Axonometric projection
• definition, point, line, plane.

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Line Plane
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Important Points of a Line

• Trace points P, N, M.
• Pp?p(x,y) zP0
• Np?n(x,z) yN0
• Mp?m(y,z) xM0

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Important Lines of a Plane

• Traces pa, na , ma
• pa a?p(x,y),
• na a?n(x,z),
• maa?m(y,z).
• Horizontal lines jh of a plane a are lines
  parallel to p(x,y).
• Frontal lines jf of a plane a are lines parallel
  to n(x,z).
• Third principal lines of a plane a are lines
  parallel to ?(y,z).

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Orthographic Projection - features

• Orthographic projection is a parallel projection
  ?It has all features of the parallel projection.

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Preservation of the Right Angle

• Theorem
• Let a, b a pair of perpendicular straight lines
  (neither of them perpendicular to ?). Projections
  of a, b are perpendicular if and only if at least
  one of the lines a, b is parallel to the image
  plane.

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Line Perpendicular to a Plane

• Theorem
• If a line k is perpendicular to a plane a then
• k1 is perpendicular to the trace p1a (and to all
  horizontals jh1),
• k2 is perpendicular to the trace n2a (and to all
  frontals jf1).

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Orthographic Projection onto Two Planes (Monge
Projection)
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Monge Projection a point
2D projection of point A

• 3D - point A in space.

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Monge Projection a line
2D projection of line p

• 3D - line p in space.

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Monge Projection a plane
2D projection of plane ?

• 3D plane ? in space.

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Monge Projection

• Ex1 Projection of a cube ABCDEFGH
• Ex2 Sketch projection of a cylinder, a cone, a
  pyramid, a sphere in Monge projection.

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Application of the principal lines

• HW Draw an unfoldment (development) of a part of
  the given prismatic surface between ?(x,y) and
  the given plane ?.

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Axonometric Projection
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Axonometry - definition

• Let S O OA, OB, OC be a Cartesian system of
  coordinates in the space. Let (?,s) be a parallel
  projection such that the projection of no axes is
  a point. Images of the coordinate axes x, y, z
  are three mutually different lines xa, ya, za
  going through one common point Oa the image of
  the origin O. Such projection is called
  axonometric projection axonometry.

O origin OA, OB, OC unit segments on axes jx,
jy, jz axonometric units
? axonometric image plane s direction of
parallel projection
Unique determination of the axonometry by five
parameters jx, jy, jz, lt(x, z), lt(y, z)
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Classification of the Axonomtery

• According to size of the axonometric units
• isometry jx jy jz
• dimetry jx jy or jz jy or jx jz
• trimetry jx? jy? jz
• According to direction of the projection
• orthographic
• oblique / general

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Axonometry - uniqueness

• describes an 3D object in 2D by means of a pair
  of parallel projections
• 1) parallel projection of the object into r (that
  is your sheet of paper)
• 2) parallel projection of either the top view or
  the front view or the side view of the object
  into r

The coordinate system is ALWAYS projected
together with the object!
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Axonometry of a Point
Unique representation of the point M M,M1,
MM1z or M,M2, MM2y or M,M3, MM3x.

• Real size of coordinates xM, yM, zM
• Axonometric units (distortion of units)
• jx, jy, jz

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Axonometry of a Line

• Important points of a line trace points P, N,
  M.
• Pp?p(x,y) zP0
• Np?n(x,z) yN0
• Mp?m(y,z) xM0
• Unique representation
• of the line p
• p,p1 or
• p,p2 or
• p,p3.

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Axonometry of a Plane

• Traces pa, na, ma
• pa a?p(x,y),
• na a?n(x,z),
• maa?m(y,z).

Traces pa, na, ma intersect on the coordinate
axes or each two are parallel.
Parallel planes have parallel traces.
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Next time

• oblique projection.