REFLECTIONS AND SHEARS

REFLECTIONS AND SHEARS

ADDED TRANSFORMATIONS

In addition to translation, rotation, and scaling, there are various additional transformations that are often useful in three-dimensional graphics applications. Two of those are reflection and shear.

Reflections

A three-dimensional reflection is performed relative to a particular reflection axis or with regard to a particular reflection plane. In general, three-dimensional reflection matrices are founded similarly to those for 2 dimensions. Reflections relative to a given axis are resembling 180 degree rotations that axis. Reflections with regard to a plane are resembling 180 degree rotations in four-dimensional space. When the reflection plane could be a coordinate plane (either xy, xz, or yz), we will consider the transformation as a conversion between Left-handed and right-handed systems. An example of a mirrored image that converts coordinate specifications from a right-handed system to a left-handed system (or vice versa). This transformation changes the sign of the z coordinates, leaving the x- and y-coordinate values unchanged. The matrix representation for this reflection of points relative to the xy plane is

Transformation matrices for inverting x and y values are defined similarly, as reflections relative to the yz plane and xz plane, respectively Reflections about other planes is obtained as a mix of rotations and coordinate-plane reflections.

Shears

Shearing transformations can he wont to modify object shapes. they're also useful in three-dimensional viewing for obtaining general projection transformations. In two dimensions, we discussed transformations relative to the x or y axes to supply distortions within the shapes of objects. In three dimensions, we will also generate shears relative to the z axis. As an example of three-dimensional shearing.

Parameters a and b is assigned any real values. The effect of this transformation matrix is to change x- and y-coordinate values by an amount that's proportional to the two value, while leaving the z coordinate unchanged. Boundaries of planes that are perpendicular to the z axis are thus shifted by an amount proportional to z.

Comments

REFLECTIONS AND SHEARS

ADDED TRANSFORMATIONS

In addition to translation, rotation, and scaling, there are various additional transformations that are often useful in three-dimensional graphics applications. Two of those are reflection and shear.

Reflections

A three-dimensional reflection is performed relative to a particular reflection axis or with regard to a particular reflection plane. In general, three-dimensional reflection matrices are founded similarly to those for 2 dimensions. Reflections relative to a given axis are resembling 180 degree rotations that axis. Reflections with regard to a plane are resembling 180 degree rotations in four-dimensional space. When the reflection plane could be a coordinate plane (either xy, xz, or yz), we will consider the transformation as a conversion between Left-handed and right-handed systems. An example of a mirrored image that converts coordinate specifications from a right-handed system to a left-handed system (or vice versa). This transformation changes the sign of the z coordinates, leaving the x- and y-coordinate values unchanged. The matrix representation for this reflection of points relative to the xy plane is

Transformation matrices for inverting x and y values are defined similarly, as reflections relative to the yz plane and xz plane, respectively Reflections about other planes is obtained as a mix of rotations and coordinate-plane reflections.

Shears

Shearing transformations can he wont to modify object shapes. they're also useful in three-dimensional viewing for obtaining general projection transformations. In two dimensions, we discussed transformations relative to the x or y axes to supply distortions within the shapes of objects. In three dimensions, we will also generate shears relative to the z axis. As an example of three-dimensional shearing.

Parameters a and b is assigned any real values. The effect of this transformation matrix is to change x- and y-coordinate values by an amount that's proportional to the two value, while leaving the z coordinate unchanged. Boundaries of planes that are perpendicular to the z axis are thus shifted by an amount proportional to z.

Comments

REFLECTIONS AND SHEARS

ADDED TRANSFORMATIONS

In addition to translation, rotation, and scaling, there are various additional transformations that are often useful in three-dimensional graphics applications. Two of those are reflection and shear.

Reflections

A three-dimensional reflection is performed relative to a particular reflection axis or with regard to a particular reflection plane. In general, three-dimensional reflection matrices are founded similarly to those for 2 dimensions. Reflections relative to a given axis are resembling 180 degree rotations that axis. Reflections with regard to a plane are resembling 180 degree rotations in four-dimensional space. When the reflection plane could be a coordinate plane (either xy, xz, or yz), we will consider the transformation as a conversion between Left-handed and right-handed systems. An example of a mirrored image that converts coordinate specifications from a right-handed system to a left-handed system (or vice versa). This transformation changes the sign of the z coordinates, leaving the x- and y-coordinate values unchanged. The matrix representation for this reflection of points relative to the xy plane is

Transformation matrices for inverting x and y values are defined similarly, as reflections relative to the yz plane and xz plane, respectively Reflections about other planes is obtained as a mix of rotations and coordinate-plane reflections.

Shears

Shearing transformations can he wont to modify object shapes. they're also useful in three-dimensional viewing for obtaining general projection transformations. In two dimensions, we discussed transformations relative to the x or y axes to supply distortions within the shapes of objects. In three dimensions, we will also generate shears relative to the z axis. As an example of three-dimensional shearing.

Parameters a and b is assigned any real values. The effect of this transformation matrix is to change x- and y-coordinate values by an amount that's proportional to the two value, while leaving the z coordinate unchanged. Boundaries of planes that are perpendicular to the z axis are thus shifted by an amount proportional to z.

Comments

REFLECTIONS AND SHEARS

ADDED TRANSFORMATIONS

In addition to translation, rotation, and scaling, there are various additional transformations that are often useful in three-dimensional graphics applications. Two of those are reflection and shear.

Reflections

A three-dimensional reflection is performed relative to a particular reflection axis or with regard to a particular reflection plane. In general, three-dimensional reflection matrices are founded similarly to those for 2 dimensions. Reflections relative to a given axis are resembling 180 degree rotations that axis. Reflections with regard to a plane are resembling 180 degree rotations in four-dimensional space. When the reflection plane could be a coordinate plane (either xy, xz, or yz), we will consider the transformation as a conversion between Left-handed and right-handed systems. An example of a mirrored image that converts coordinate specifications from a right-handed system to a left-handed system (or vice versa). This transformation changes the sign of the z coordinates, leaving the x- and y-coordinate values unchanged. The matrix representation for this reflection of points relative to the xy plane is

Transformation matrices for inverting x and y values are defined similarly, as reflections relative to the yz plane and xz plane, respectively Reflections about other planes is obtained as a mix of rotations and coordinate-plane reflections.

Shears

Shearing transformations can he wont to modify object shapes. they're also useful in three-dimensional viewing for obtaining general projection transformations. In two dimensions, we discussed transformations relative to the x or y axes to supply distortions within the shapes of objects. In three dimensions, we will also generate shears relative to the z axis. As an example of three-dimensional shearing.

Parameters a and b is assigned any real values. The effect of this transformation matrix is to change x- and y-coordinate values by an amount that's proportional to the two value, while leaving the z coordinate unchanged. Boundaries of planes that are perpendicular to the z axis are thus shifted by an amount proportional to z.

Comments

REFLECTIONS AND SHEARS

ADDED TRANSFORMATIONS

In addition to translation, rotation, and scaling, there are various additional transformations that are often useful in three-dimensional graphics applications. Two of those are reflection and shear.

Reflections

A three-dimensional reflection is performed relative to a particular reflection axis or with regard to a particular reflection plane. In general, three-dimensional reflection matrices are founded similarly to those for 2 dimensions. Reflections relative to a given axis are resembling 180 degree rotations that axis. Reflections with regard to a plane are resembling 180 degree rotations in four-dimensional space. When the reflection plane could be a coordinate plane (either xy, xz, or yz), we will consider the transformation as a conversion between Left-handed and right-handed systems. An example of a mirrored image that converts coordinate specifications from a right-handed system to a left-handed system (or vice versa). This transformation changes the sign of the z coordinates, leaving the x- and y-coordinate values unchanged. The matrix representation for this reflection of points relative to the xy plane is

Transformation matrices for inverting x and y values are defined similarly, as reflections relative to the yz plane and xz plane, respectively Reflections about other planes is obtained as a mix of rotations and coordinate-plane reflections.

Shears

Shearing transformations can he wont to modify object shapes. they're also useful in three-dimensional viewing for obtaining general projection transformations. In two dimensions, we discussed transformations relative to the x or y axes to supply distortions within the shapes of objects. In three dimensions, we will also generate shears relative to the z axis. As an example of three-dimensional shearing.

Parameters a and b is assigned any real values. The effect of this transformation matrix is to change x- and y-coordinate values by an amount that's proportional to the two value, while leaving the z coordinate unchanged. Boundaries of planes that are perpendicular to the z axis are thus shifted by an amount proportional to z.

Comments