Universitat Polit ecnica de Catalunya Facultat d’Inform ... · Computational Geometry, Facultat...

Rodrigo Silveira

Computational Geometry

Facultat d’Informatica de Barcelona

Universitat Politecnica de Catalunya

Basic tools:orientation tests

Turning right or turning left?

Computational Geometry, Facultat d’Informatica de Barcelona, UPC

Consider a sequence of 3 points: p,q, r

pq

r



Basic question: do they make a right turn?A left turn?Or perhaps they are colinear?


pq

r





pq

r

This is a fundamental basic operation in geometric algorithms. Requires a solution that is

• Efficient

• Numerically robust





pq

r

This is a fundamental basic operation in geometric algorithms. Requires a solution that is

• Efficient

• Numerically robust

Solution: signed area of triangle 4pqr

Signed area of triangle


The signed area of 4pqr tells us the area ofthe triangle, and the sign gives the orientation(left-turning or right-turning)

Signed area of 4pqr

pq

r

• Can the area be zero?



The signed area of 4pqr tells us the area ofthe triangle, and the sign gives the orientation(left-turning or right-turning)

Signed area of 4pqr

pq

r

• A(4pqr) is twice the area of the parallelogramdefined by vectors pq and w

• Recall: in 2-dimensions, the area of the paral-lelogram defined by two vectors is the deter-minant of a 2 × 2 matrix whose columns arethe two vectors

• Can the area be zero?

Computing A(4pqr)

pq

rw = r − p

pq = q − p



Signed area of 4pqr

A(4pqr) =1

2

∣∣∣∣∣ pqx wx

pqy wy

∣∣∣∣∣

p

q

rw = r − p

pq = q − p



Signed area of 4pqr

A(4pqr) =1

2


pqy wy

∣∣∣∣∣

p

q

rw = r − p

pq = q − p

=1

2

∣∣∣∣∣ (q − p)x (r − p)x

(q − p)y (r − p)y

∣∣∣∣∣=

1

2

∣∣∣∣∣ qx − px rx − px

qy − py ry − py

∣∣∣∣∣=

1

2

∣∣∣∣∣∣∣∣px qx rx

py qy ry

1 1 1

∣∣∣∣∣∣∣∣



Signed area of 4pqr

A(4pqr) =1

2


pqy wy

∣∣∣∣∣

p

q

rw = r − p

pq = q − p

=1

2

∣∣∣∣∣ (q − p)x (r − p)x

(q − p)y (r − p)y

∣∣∣∣∣=

1

2

∣∣∣∣∣ qx − px rx − px

qy − py ry − py

∣∣∣∣∣=

1

2

∣∣∣∣∣∣∣∣px qx rx

py qy ry

1 1 1

∣∣∣∣∣∣∣∣Fact: (check it yourself!)

A(4pqr) > 0 if and only if p, q, r are in counter-clockwise order

Does it turn right or left?

More things you can do with this test


Is point r to the right or left of the oriented line `?

Point with respect to a line

r

p

q

`

RIGHT

LEFT





r

p

q

`

Equivalent question: do p, q, r turn right or left?

RIGHT

LEFT





r

p

q

`

Line-segment intersection test


`

pq

r

sDoes segment rs intersect line `?

RIGHT

LEFT

RIGHT

LEFT





r

p

q

`

Line-segment intersection test


`

pq

r

sDoes segment rs intersect line `?

rs intersects line ` ↔ r and s are on opposite sides of `

RIGHT

LEFT

RIGHT

LEFT

↔ (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 `)

rs intersects line ` ↔ r and s are on opposite sides of `



Does segment rs intersect halfline h?

Halfline-segment intersection

r

p

h

RIGHT

LEFT

q

s





r

p

h

Equivalent question(s):

• r and s are on opposite sides of h(so we can assume r is to the left of h)

RIGHT

LEFT

q

s





r

p

h



RIGHT

LEFT

q

s

hr

s

p

RIGHTLEFT

q





r

p

h



RIGHT

LEFT

q

s

hr

s

p

RIGHTLEFT

qp h

q

r

sRIGHTLEFT

ph

q

r

sRIGHTLEFT





r

p

h



RIGHT

LEFT

q

s

• r, s, p makes a right turn

hr

s

p

RIGHTLEFT

qp h

q

r

sRIGHTLEFT

ph

q

r

sRIGHTLEFT


Does segment pq intersect segment rs?

Segment-segment intersection

r

p

q

s





r

p

q

s

r

p

q

s





r

p


• r and s are on opposite “sides” of pq

q

s

r

p

q

s

Computational Geometry, Facultat d’Informatica de Barcelona, UPCComputational Geometry, Facultat d’Informatica de Barcelona, UPCComputational Geometry, Facultat d’Informatica de Barcelona, UPC




r

p


• r and s are on opposite “sides” of pq

q

s

• p and q are on opposite “sides” of rs

r

p

q

s


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!

Summary









Another useful test: point in circle• See notes on course webpage

Summary









Questions?

Another useful test: point in circle• See notes on course webpage

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