Basic tools:
orientation tests
Rodrigo Silveira
Computational Geometry
Facultat d’Informàtica de Barcelona
Universitat Politècnica de Catalunya
Turning right or turning left?
Consider a sequence of 3 points: p, q, r
r
p
q
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Turning right or turning left?
Consider a sequence of 3 points: p, q, r
r
p
q
Basic question: do they make a right turn?
A left turn?
Or perhaps they are colinear?
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Turning right or turning left?
Consider a sequence of 3 points: p, q, r
r
p
q
Basic question: do they make a right turn?
A left turn?
Or perhaps they are colinear?
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Turning right or turning left?
Consider a sequence of 3 points: p, q, r
r
p
q
Basic question: do they make a right turn?
A left turn?
Or perhaps they are colinear?
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Turning right or turning left?
Consider a sequence of 3 points: p, q, r
r
p
q
Basic question: do they make a right turn?
A left turn?
Or perhaps they are colinear?
This is a fundamental basic operation in geometric algorithms. Requires a solution that is
• Efficient
• Numerically robust
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Turning right or turning left?
Consider a sequence of 3 points: p, q, r
r
p
q
Basic question: do they make a right turn?
A left turn?
Or perhaps they are colinear?
This is a fundamental basic operation in geometric algorithms. Requires a solution that is
• Efficient
• Numerically robust
Solution: signed area of triangle 4pqr
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Signed area of triangle
Signed area of 4pqr
r
The signed area of 4pqr tells us the area of
the triangle, and the sign gives the orientation
(left-turning or right-turning)
p
q
• Can the area be zero?
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Signed area of triangle
Signed area of 4pqr
r
The signed area of 4pqr tells us the area of
the triangle, and the sign gives the orientation
(left-turning or right-turning)
p
q
• Can the area be zero?
Computing A(4pqr)
• A(4pqr) is twice the area of the parallelogram
defined by vectors pq and w
• Recall: in 2-dimensions, the area of the parallelogram defined by two vectors is the determinant of a 2 × 2 matrix whose columns are
the two vectors
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
w =r−p
r
p
pq = q − p
q
Signed area of triangle
Signed area of 4pqr
1 pq x
A(4pqr) = 2 pq y
wx
wy
w =r−p
r
p
pq = q − p
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
q
Signed area of triangle
Signed area of 4pqr
1 A(4pqr) = 2
1 = 2
1 = 2
1 = 2
pq x
wx
pq y
wy
(q − p)x
(q − p)y
qx − px
qy − py
px
qx
py
qy
1
1
w =r−p
(r − p)x (r − p)y rx − px ry − py rx ry 1 Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
r
p
pq = q − p
q
Signed area of triangle
Signed area of 4pqr
1 A(4pqr) = 2
1 = 2
1 = 2
1 = 2
pq x
wx
pq y
wy
(q − p)x
(q − p)y
qx − px
qy − py
px
qx
py
qy
1
1
w =r−p
(r − p)x (r − p)y rx − px ry − py rx ry 1 r
p
pq = q − p
Does it turn right or left?
Fact: (check it yourself!)
A(4pqr) > 0 if and only if p, q, r are in counter-clockwise order
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
q
More things you can do with this test
Point with respect to a line
LEFT
r
q
`
RIGHT
Is point r to the right or left of the oriented line `?
p
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Point with respect to a line
LEFT
r
q
`
RIGHT
Is point r to the right or left of the oriented line `?
p
Equivalent question: do p, q, r turn right or left?
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Point with respect to a line
LEFT
r
q
`
RIGHT
Is point r to the right or left of the oriented line `?
p
Equivalent question: do p, q, r turn right or left?
Line-segment intersection test
r
`
p
Does segment rs intersect line `?
q
s
LEFT
RIGHT
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Point with respect to a line
LEFT
r
q
`
RIGHT
Is point r to the right or left of the oriented line `?
p
Equivalent question: do p, q, r turn right or left?
Line-segment intersection test
r
`
p
Does segment rs intersect line `?
q
s
rs intersects line ` ↔ r and s are on opposite sides of `
LEFT
RIGHT
↔ (r is to the left and s is to the right of `) OR (r is to the right and s is to the left of `)
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Halfline-segment intersection
LEFT
RIGHT
r
h
q
Does segment rs intersect halfline h?
p
s
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Halfline-segment intersection
LEFT
RIGHT
r
h
q
Does segment rs intersect halfline h?
p
Equivalent question(s):
s
• r and s are on opposite sides of h
(so we can assume r is to the left of h)
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Halfline-segment intersection
LEFT
RIGHT
r
h
q
Does segment rs intersect halfline h?
p
Equivalent question(s):
• r and s are on opposite sides of h
(so we can assume r is to the left of h)
s
r
p
h
q
LEFT
RIGHT s
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Halfline-segment intersection
LEFT
RIGHT
r
h
q
Does segment rs intersect halfline h?
p
Equivalent question(s):
• r and s are on opposite sides of h
(so we can assume r is to the left of h)
s
r
p
LEFT
RIGHT s
h
q
r
h
p
RIGHT
q
LEFT
s
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
r
h
p
LEFT
s
RIGHT
q
More things you can do with this test
Halfline-segment intersection
LEFT
RIGHT
r
h
q
Does segment rs intersect halfline h?
p
Equivalent question(s):
• r and s are on opposite sides of h
(so we can assume r is to the left of h)
• r, s, p makes a right turn
s
r
p
LEFT
RIGHT s
h
q
r
h
p
RIGHT
q
LEFT
s
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
r
h
p
LEFT
s
RIGHT
q
More things you can do with this test
Segment-segment intersection
Does segment pq intersect segment rs?
r
q
p
s
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Segment-segment intersection
Does segment pq intersect segment rs?
r
q
p
r
q
s
p
s
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Segment-segment intersection
Does segment pq intersect segment rs?
r
q
p
r
q
s
p
s
Equivalent question(s):
• r and s are on opposite “sides” of pq
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Segment-segment intersection
Does segment pq intersect segment rs?
r
q
p
r
q
s
p
s
Equivalent question(s):
• r and s are on opposite “sides” of pq
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
More things you can do with this test
Segment-segment intersection
Does segment pq intersect segment rs?
r
q
p
r
q
s
p
s
Equivalent question(s):
• r and s are on opposite “sides” of pq
• p and q are on opposite “sides” of rs
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Summary
Things we can do with the orientation test
• Test if p, q, r make a right turn
• Test if a point is to the left of a line
• Test if a segment and a line intersect
• Test if a segment and a halfline intersect
• Test if two segments intersect
• Test if a point is inside a triangle
All by evaluating the sign of a couple of determinants!
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Summary
Things we can do with the orientation test
• Test if p, q, r make a right turn
• Test if a point is to the left of a line
• Test if a segment and a line intersect
• Test if a segment and a halfline intersect
• Test if two segments intersect
• Test if a point is inside a triangle
All by evaluating the sign of a couple of determinants!
Another useful test: point in circle
• See notes on course webpage
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC
Summary
Things we can do with the orientation test
• Test if p, q, r make a right turn
• Test if a point is to the left of a line
• Test if a segment and a line intersect
• Test if a segment and a halfline intersect
• Test if two segments intersect
• Test if a point is inside a triangle
All by evaluating the sign of a couple of determinants!
Another useful test: point in circle
• See notes on course webpage
Questions?
Computational Geometry, Facultat d’Informàtica de Barcelona, UPC