Point Classes

import introcs

Points have position, but they do not have magnitude or direction. Use the class Vector if you want direction. Points support basic point arithmetic via the operators. However, pay close attention to how we handle typing. For example, the difference between two points is a vector (as it should be). But points may freely convert to vectors and vice versa.

Class Point2

This class provides 2-dimensional points. It is the primary geometry class used by the game2d package.

Constructor

class introcs.Point2(x=0, y=0)

An instance is a point in 2D space.

All attribute values are 0.0 by default.

Parameters:

  • x (int or float) – initial x value

  • y (int or float) – initial y value

Attributes

Point2.x

The x coordinate

Invariant: Value must be an int or float.

Point2.y

The y coordinate

Invariant: Value must be an int or float.

Immutable Methods

Immutable methods return a new object and do not modify the original.

Point2.toVector()
Returns:

The Vector2 object equivalent to this point

Return type:

Vector2

Point2.midpoint(other)

Computes the midpoint between self and other.

This method treats self and other as a line segment, so they must both be points.

Parameters:

other (Point2) – the other end of the line segment

Returns:

the midpoint between this point and other

Return type:

Point2

Point2.distance(other)

Computes the Euclidean between two points

Parameters:

other (Point2) – value to compare against

Returns:

the Euclidean distance from this point to other

Return type:

float

Point2.distance2(other)

Computes the squared Euclidean between two points

This method is slightly faster than distance().

Parameters:

other (Point2) – value to compare against

Returns:

the squared Euclidean distance from this point to other

Return type:

float

Point2.under(other)

Compares self to other under the domination partial order

We say that one point dominates another is all components of the first are greater than or equal to the components of the second. This is a partial order, not a total one.

Parameters:

other (Vector2) – The object to check

Returns:

True if other dominates self; False otherwise

Return type:

bool

Point2.over(other)

Compares self to other under the domination partial order

We say that one point dominates another is all components of the first are greater than or equal to the components of the second. This is a partial order, not a total one.

Parameters:

other (Vector2) – The object to check

Returns:

True if self dominates other; False otherwise

Return type:

bool

Point2.isZero()

Determines whether or not this object is ‘close enough’ to the origin.

This method uses allclose() to test whether the coordinates are “close enough”. It does not require exact equality for floats.

Returns:

True if this object is ‘close enough’ to the origin; False otherwise

Return type:

bool

Point2.interpolant(other, alpha)

Interpolates this object with another, producing a new object

The resulting value is:

alpha*self+(1-alpha)*other

according to the rules of addition and scalar multiplication.

Parameters:

  • other (Vector2) – object to interpolate with

  • alpha (int or float) – scalar to interpolate by

Returns:

the interpolation of this object and other via alpha.

Return type:

Vector2

Point2.copy()
Returns:

A copy of this point

Return type:

Vector2

Point2.list()
Returns:

A python list with the contents of this point.

Return type:

list

Mutable Methods

Mutable methods modify the underlying object.

Point2.interpolate(other, alpha)

Interpolates this object with another in place

This method will modify the attributes of this oject. The new attributes will be equivalent to:

alpha*self+(1-alpha)*other

according to the rules of addition and scalar multiplication.

This method returns this object for chaining.

Parameters:

  • other (Vector2) – object to interpolate with

  • alpha (int or float) – scalar to interpolate by

Returns:

This object, newly modified

Point2.clamp(low, high)

Clamps this point to the range [low, high].

Any value in this tuple less than low is set to low. Any value greater than high is set to high.

This method returns this object for chaining.

Parameters:

  • low (int or float) – The low range of the clamp

  • high (int or float) – The high range of the clamp

Returns:

This object, newly modified

Return type:

Vector2

Operators

Operators redefine the meaning of the basic operations. For example:: p + q is the same as p.__add__(q). This allows us to treat points like regular numbers. For the sake of brevity, we have not listed all operators – only the most important ones. The equivalences are as follows:

p == q     -->    p.__eq__(q)
p < q      -->    p.__lt__(q)
p + q      -->    p.__add__(q)
p - q      -->    p.__sub__(q)
p * q      -->    p.__mul__(q)
q * p      -->    p.__rmul__(q)
p / q      -->    p.__truediv__(q)
q / p      -->    p.__rtruediv__(q)
Point2.__eq__(other)

Compares this point with other

This method uses allclose() to test whether the coordinates are “close enough”. It does not require exact equality for floats. Equivalence also requires type equivalence.

Parameters:

other (any) – The object to check

Returns:

True if self and other are equivalent

Return type:

bool

Point2.__lt__(other)

Compares the lexicographic ordering of self and other.

Lexicographic ordering checks the x-coordinate first, and then y.

Parameters:

other (Vector2) – The object to check

Returns:

True if self is lexicographic kess than other

Return type:

float

Point2.__add__(other)

Performs a context dependent addition of this point and other.

If other is a point, the result is the vector from this position to other (so other is the head). If it is a vector, it is the point at the head of the vector when it is anchored at this point.

Parameters:

other (Point2 or Vector2) – object to add

Returns:

the sum of this object and other.

Return type:

Point2 or Vector2

Point2.__sub__(other)

Performs a context dependent subtraction of this point and other.

If other is a point, the result is the vector from other to this position (so other is the tail). If it is a vector, it is the point at the tail of the vector whose head is at this point.

Parameters:

other (Point2 or Vector2) – object to subtract

Returns:

the difference of this object and other.

Return type:

Point2 or Vector2

Point2.__mul__(value)

Multiples this object by a scalar, Vector2, or a Matrix, producing a new object.

The exact effect is determined by the type of value. If value is a scalar, the result is standard scalar multiplication. If it is a point, then the result is pointwise multiplication. Finally, if is a matrix, then we use the matrix to transform the object. We treat matrix transformation as multiplication on the right to make in-place multiplication easier. See Matrix doe more

Parameters:

value (int, float, Vector2 or Matrix) – value to multiply by

Returns:

the altered object

Return type:

Vector2

Point2.__rmul__(value)

Multiplies this object by a scalar or Vector2 on the left.

The exact effect is determined by the type of value. If value is a scalar, the result is standard scalar multiplication. If it is a 2d tuple, then the result is pointwise multiplication. We do not allow matrix multiplication on the left.

Parameters:

value (int, float, or Vector2) – The value to multiply by

Returns:

the scalar multiple of self and scalar

Return type:

Vector2

Point2.__truediv__(value)

Divides this object by a scalar or a Vector2 on the right, producting a new object.

The exact effect is determined by the type of value. If value is a scalar, the result is standard scalar division. If it is a Vector2, then the result is pointwise division.

The value returned has the same type as self (so if self is an instance of a subclass, it uses that object instead of the original class. The contents of this object are not altered.

Parameters:

value (int, float, or Vector2) – The value to multiply by

Returns:

the division of self by value

Return type:

Vector2

Point2.__rtruediv__(value)

Divides a scalar or Vector2 by this object.

Dividing by a point means pointwise reciprocation, followed by multiplication.

Parameters:

value (int, float, or Vector2) – The value to divide

Returns:

the division of value by self

Return type:

Class Point3

This class provides 3-dimensional points. It will not be used by this assignment. However, the name Point is an alias for Point3.

Constructor

class introcs.Point3(x=0, y=0, z=0)

An instance is a point in 3D space.

All attribute values are 0.0 by default.

Parameters:

  • x (int or float) – initial x value

  • y (int or float) – initial y value

  • z (int or float) – initial z value

Attributes

Point3.x

The x coordinate

Invariant: Value must be an int or float.

Point3.y

The y coordinate

Invariant: Value must be an int or float.

Immutable Methods

Immutable methods return a new object and do not modify the original.

Point3.toVector()
Returns:

The Vector3 object equivalent to this point

Return type:

Vector3

Point3.midpoint(other)

Computes the midpoint between self and other.

This method treats self and other as a line segment, so they must both be points.

Parameters:

other (Point3) – the other end of the line segment

Returns:

the midpoint between this point and other

Return type:

Point3

Point3.distance(other)

Computes the Euclidean between two points

Parameters:

other (Point3) – value to compare against

Returns:

the Euclidean distance from this point to other

Return type:

float

Point3.distance2(other)

Computes the squared Euclidean between two points

This method is slightly faster than distance().

Parameters:

other (Point3) – value to compare against

Returns:

the squared Euclidean distance from this point to other

Return type:

float

Point3.under(other)

Compares self to other under the domination partial order

We say that one point dominates another is all components of the first are greater than or equal to the components of the second. This is a partial order, not a total one.

Parameters:

other (Vector3) – The object to check

Returns:

True if other dominates self; False otherwise

Return type:

bool

Point3.over(other)

Compares self to other under the domination partial order

We say that one point dominates another is all components of the first are greater than or equal to the components of the second. This is a partial order, not a total one.

Parameters:

other (Vector3) – The object to check

Returns:

True if self dominates other; False otherwise

Return type:

bool

Point3.isZero()

Determines whether or not this object is ‘close enough’ to the origin.

This method uses allclose() to test whether the coordinates are “close enough”. It does not require exact equality for floats.

Returns:

True if this object is ‘close enough’ to the origin; False otherwise

Return type:

bool

Point3.interpolant(other, alpha)

Interpolates this object with another, producing a new object

The resulting value is:

alpha*self+(1-alpha)*other

according to the rules of addition and scalar multiplication.

Parameters:

  • other (Vector3) – object to interpolate with

  • alpha (int or float) – scalar to interpolate by

Returns:

the interpolation of this object and other via alpha.

Return type:

Vector3

Point3.copy()
Returns:

A copy of this point

Return type:

Vector3

Point3.list()
Returns:

A python list with the contents of this point.

Return type:

list

Mutable Methods

Mutable methods modify the underlying object.

Point3.interpolate(other, alpha)

Interpolates this object with another in place

This method will modify the attributes of this oject. The new attributes will be equivalent to:

alpha*self+(1-alpha)*other

according to the rules of addition and scalar multiplication.

This method returns this object for chaining.

Parameters:

  • other (Vector3) – object to interpolate with

  • alpha (int or float) – scalar to interpolate by

Returns:

This object, newly modified

Point3.clamp(low, high)

Clamps this point to the range [low, high].

Any value in this tuple less than low is set to low. Any value greater than high is set to high.

This method returns this object for chaining.

Parameters:

  • low (int or float) – The low range of the clamp

  • high (int or float) – The high range of the clamp

Returns:

This object, newly modified

Return type:

Vector3

Operators

Operators redefine the meaning of the basic operations. For example:: p + q is the same as p.__add__(q). This allows us to treat points like regular numbers. For the sake of brevity, we have not listed all operators – only the most important ones. The equivalences are as follows:

p == q     -->    p.__eq__(q)
p < q      -->    p.__lt__(q)
p + q      -->    p.__add__(q)
p - q      -->    p.__sub__(q)
p * q      -->    p.__mul__(q)
q * p      -->    p.__rmul__(q)
p / q      -->    p.__truediv__(q)
q / p      -->    p.__rtruediv__(q)
Point3.__eq__(other)

Compares this point with other

This method uses allclose() to test whether the coordinates are “close enough”. It does not require exact equality for floats. Equivalence also requires type equivalence.

Parameters:

other (any) – The object to check

Returns:

True if self and other are equivalent

Return type:

bool

Point3.__lt__(other)

Compares the lexicographic ordering of self and other.

Lexicographic ordering checks the x-coordinate first, and then y.

Parameters:

other (Vector3) – The object to check

Returns:

True if self is lexicographic kess than other

Return type:

float

Point3.__add__(other)

Performs a context dependent addition of this point and other.

If other is a point, the result is the vector from this position to other (so other is the head). If it is a vector, it is the point at the head of the vector when it is anchored at this point.

Parameters:

other (Point2 or Vector2) – object to add

Returns:

the sum of this object and other.

Return type:

Point2 or Vector2

Point3.__sub__(other)

Performs a context dependent subtraction of this point and other.

If other is a point, the result is the vector from other to this position (so other is the tail). If it is a vector, it is the point at the tail of the vector whose head is at this point.

Parameters:

other (Point3 or Vector3) – object to subtract

Returns:

the difference of this object and other.

Return type:

Point3 or Vector3

Point3.__mul__(value)

Multiples this object by a scalar, Vector3, or a Matrix, producing a new object.

The exact effect is determined by the type of value. If value is a scalar, the result is standard scalar multiplication. If it is a point, then the result is pointwise multiplication. Finally, if is a matrix, then we use the matrix to transform the object. We treat matrix transformation as multiplication on the right to make in-place multiplication easier. See Matrix doe more

Parameters:

value (int, float, Vector3 or Matrix) – value to multiply by

Returns:

the altered object

Return type:

Vector3

Point3.__rmul__(value)

Multiplies this object by a scalar or Vector3 on the left.

The exact effect is determined by the type of value. If value is a scalar, the result is standard scalar multiplication. If it is a 2d tuple, then the result is pointwise multiplication. We do not allow matrix multiplication on the left.

Parameters:

value (int, float, or Vector3) – The value to multiply by

Returns:

the scalar multiple of self and scalar

Return type:

Vector3

Point3.__truediv__(value)

Divides this object by a scalar or a Vector3 on the right, producting a new object.

The exact effect is determined by the type of value. If value is a scalar, the result is standard scalar division. If it is a Vector3, then the result is pointwise division.

The value returned has the same type as self (so if self is an instance of a subclass, it uses that object instead of the original class. The contents of this object are not altered.

Parameters:

value (int, float, or Vector3) – The value to multiply by

Returns:

the division of self by value

Return type:

Vector3

Point3.__rtruediv__(value)

Divides a scalar or Vector3 by this object.

Dividing by a point means pointwise reciprocation, followed by multiplication.

Parameters:

value (int, float, or Vector3) – The value to divide

Returns:

the division of value by self

Return type: