# Point3¶

`import introcs`

Points have position, but they do not have magnitude or direction. Use the class `Vector3` 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.

The name `Point` is an alias for `Point3`.

## Constructor¶

class `introcs.``Point3`([x, [y, [z]]])

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.__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