Points and Vectors

Points and vectors are a fundamental geometric primitive within the engeom library that allow for the representation of positions and directions in 2D and 3D space. They are used as components in a number of other geometric entities, but can also be used directly.

Note

When working with large numbers of points or vectors in Python, it is much more efficient to use numpy arrays than it is to use Python lists of Point2/Point3 or Vector2/Vector3 objects. The engeom library provides a number of functions which can operate directly on numpy.ndarrays for clarity and speed.

The 2D point and vector types are located in the geom2 module, while the 3D point and vector types are located in the geom3 module.

from engeom.geom2 import Point2, Vector2
from engeom.geom3 import Point3, Vector3

Creation

Points and vectors both are created by specifying x, y, and (in the case of the 3D versions) z components.

from engeom.geom3 import Point3, Vector3

# Create a 3D point at (1, 2, 3)
p1 = Point3(1, 2, 3)

# Create a 3D vector with components (4, 5, 6)
v1 = Vector3(4, 5, 6)

For convenience, you can use Python's * unpacking operator to pass in a list or tuple of values.

import numpy
from engeom.geom3 import Point3, Vector3

coords = numpy.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
points = [Point3(*row) for row in coords]

named = {
    "v0": (1, 2, 3),
    "v1": (4, 5, 6),
    "v2": (7, 8, 9)
}

vec0 = Vector3(*named["v0"])
vec1 = Vector3(*named["v1"])
vec2 = Vector3(*named["v2"])

Lastly, for convenience, a point's coordinates can be extracted in the form of a vector.

from engeom.geom3 import Point3

p = Point3(1, 2, 3)

v = p.coords
print(v)  # Vector3(1, 2, 3)

Accessing Components

The x, y, and z components of a point or vector can be accessed directly.

from engeom.geom3 import Point3, Vector3

v0 = Vector3(1, 2, 3)

print(v0.x)  # 1
print(v0.y)  # 2
print(v0.z)  # 3

The components are also iterable, so you can unpack them back into a list, tuple, or into arguments for a function.

from engeom.geom3 import Point3


def some_function(a: float, b: float, c: float) -> float:
    return a + b + c


p = Point3(1, 2, 3)

x, y, z = p
coords = list(p)
value = some_function(*p)

for component in p:
    print(component)

Scaling Points and Vectors

Points and vectors can be multiplied and divided by scalars.

from engeom.geom3 import Point3, Vector3

p0 = Point3(1, 2, 3)
v0 = Vector3(4, 5, 6)

p1 = p0 * 2
p2 = 3.0 * p0
v1 = v0 / 2

Both can also be inverted, which is equivalent to multiplying by -1.

from engeom.geom3 import Point3, Vector3

p0 = Point3(1, 2, 3)
v0 = Vector3(4, 5, 6)

p1 = -p0
v1 = -v0

Adding and Subtracting Points and Vectors

Points and vectors can be added to and subtracted from each other, however, the resulting type will vary depending on the operation.

Left	Operand	Right	Result
Point	+	Vector	Point
Point	-	Vector	Point
Point	+	Point	INVALID
Point	-	Point	Vector
Vector	+	Vector	Vector
Vector	-	Vector	Vector
Vector	+	Point	INVALID
Vector	-	Point	INVALID

To summarize, a point cannot be added to anything, and a point can only be subtracted from another point. A vector can be added to or subtracted from either a point or another vector.

Vector Operations

The length of a vector can be calculated using the norm() function, and a normalized version of the vector can be created using the normalized() function.

from engeom.geom3 import Vector3

v = Vector3(1, 2, 3)

length = v.norm()
unit = v.normalized()

Vectors can be used to perform dot and cross products, and to measure the smallest angle between two vectors.

from engeom.geom3 import Vector3

v0 = Vector3(1, 2, 3)
v1 = Vector3(4, 5, 6)

d = v0.dot(v1)

angle = v0.angle_to(v1)

# The cross product will be a new vector if these are `Vector3` objects, 
# or a scalar if they are `Vector2` objects.
c = v0.cross(v1)