ORTHOGRAPHIC PROJECTION WITH ELEVATION€¦ · 17.10.2016 2 Projection of basic shapes •...
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17.10.2016 1 ORTHOGRAPHIC PROJECTION WITH ELEVATION s...........direction, s⊥π k c ..........elevation (altitude) of the point C C 1 (k c )....image of the point C. if k c >0 (k c <0), than C is up (bellow) projection plane π. determine the measure unit ( e.g. j =1cm) One projection is NOT one-to-one correspondence Image of point C∈E 3 with elevation provide 3D space information E 3 ⇔ π×R C ⇔ C 1 (k c )
Transcript of ORTHOGRAPHIC PROJECTION WITH ELEVATION€¦ · 17.10.2016 2 Projection of basic shapes •...
17.10.2016
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ORTHOGRAPHIC PROJECTION WITH ELEVATION
�s...........direction, s⊥π�kc..........elevation (altitude) of the point C�C1(kc)....image of the point C.� if kc>0 (k c<0), than C is up (bellow) projection plane π.� determine the measure unit ( e.g. j = 1cm)
�One projection is NOT one-to-one correspondence�Image of point C∈E3 with elevation provide 3D space information� E3 ⇔ π×R� C ⇔ C1(kc)
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Projection of basic shapes
• Projection of a line is a line or point.• Projection of a plane is a plane or line (edge view).• Projection of a circle – circle, ellipse, segment line.• Projection of a square is general parallelogram.
A(0)=E(1) B(0)=F(1)
C(0)=G(1)D(0)=H(1)
k(1)
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A(0)=E(1) B(0)=F(1)
C(0)=G(1)D(0)=H(1)
k(0)
EDGE VIEW – PLANE PARALLEL WITH DIRECTION
KP_ctverec:sklapeni.ggb
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TWO STRAIGHT LINES
1) Distinct but intersecting lines 2) parallels
3) Skew lines
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�Straight line is uniquely determined by two distinct points A, B�Projection of a line c perpendicular to horizontal plane π (c⊥π) is point C1
Projection of the straight line
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�PLine a ≡AB is given by two distinct points A, B�Revolve plane α to be parallel with π, - auxiliary view (A ), (B ):
|(A )A1|=|kA|, (A )A1⊥a1, |(B )B1|=|kB|, (B )B1⊥a1
Piercing point (P=a∩π ) and angle between α , π
Projection of the straight line
From equal right triangles with hypotenuses on a we get|A1X1|=|X1X2|=|X2X3|
Topographic image of line a (a ≡ AB ) is given by its points with integer elevation.
Projection of the straight line
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�Interval: i = |A1C1|, |kA-kB|=1
�Slope: s = j/i i=j/s
Projection of the straight line
Interval i distance between two neighbouring points
Slope s of a line b s = tg (α), where α is angle between b and π.
stupnovani.ggb
All points on line h have the same elevation.
Contour straight line
Horizontal line h parallel with π
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Projection of Plane
Trace p ρ of plane ρ - intersection with plane of projection.
Contour lines of plane ρ - parallel lines equally separated
Steepest line s - line of steepest descent is perpendiculat to contour lines.
Pro průměty hlavních a spádových přímek platí: h1||h||p ρ, s1⊥h1(s⊥h∧h||π)
Poznámka: Stopa roviny je hlavní přímka s nulovou kótou.
Projection of plane
Slope (gradient) of a plane ρρρρ - tg α, where α is agle between ρ and π.
α = ∠sπ = ∠ρπ
Exercise: Plane is given by its trace and the slope. Construct other contour lines.
Spad_roviny.ggb
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PROJECTION OF A CIRCLECircle k ⊂ ρ is given by centre S0 and radius r.
Plane ρ is given by contour lines.
S0 ∈ p, Major axis (unshortened) on trace p
a = rb = r cos α,
where α is slope of the plane.
KP_kruznice_otaceni.ggb
Right Circular Cone
Axis of the cone is vertical.
All straight lines on the conic surface have the same angle with horizontal plane π
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Exercise
Planes are given by steepest lines sρ, sσ. Construct their intersection line r ≡ρ∩σ
Exeercise
Construct arbitrary plane with the point V and a slope tg α = 5/4 .
spadovykuzel.ggb
