In Geometric Algebra, G(3,0,1) is a degenerate-metric algebra known as PGA, originally called Projective Geometric Algebra in prior literature. It includes within it a point-based algebra, plane-based algebra, and a dual quaternion geometric algebra (DQGA). In the point-based algebra of PGA, there are outer product null space (OPNS) geometric entities based on a 1-blade point entity, and the join (outer product) of two or three points forms a 2-blade line or 3-blade plane. In the plane-based algebra of PGA, there are commutator product null space (CPNS) geometric entities based on a 1-blade plane entity, and the meet (outer product) of two or three planes forms a 2-blade line or 3-blade point. The point-based OPNS entities are dual to the plane-based CPNS entities through a new geometric entity dualization operation J_e that is defined by careful observation of the entity duals in same orientation and collected in a table of basis-blade duals. The paper contributes the new operation J_e and its implementations using three different nondegenerate algebras {G(4),G(3,1),G(1,3)} as forms of Hodge star dualizations, which in geometric algebra are various products of entities with nondegenerate unit pseudoscalars, taking a grade k entity to its dual grade 4-k entity copied back into G(3,0,1). The paper contributes a detailed development of DQGA. DQGA represents and emulates the dual quaternion algebra (DQA) as a geometric algebra that is entirely within the even-grades subalgebra of PGA G(3,0,1). DQGA has a close relation to the plane-based CPNS PGA entities through identities, which allows to derive dual quaternion representations of points, lines, planes, and many operations on them (reflection, rotation, translation, intersection, projection), all within the dual quaternion algebra. In DQGA, all dual quaternion operations are implemented by using the larger PGA algebra. The DQGA standard operations include complex conjugate, quaternion conjugate, dual conjugate, and part operators (scalar, vector, tensor, unit, real, imaginary), and some new operations are defined for taking more parts (point, plane, line) and taking the real component of the imaginary part by using the new operation J_e. All DQGA entities and operations are derived in detail. It is possible to easily convert any point-based OPNS PGA entity to and from its dual plane-based CPNS PGA entity, and then also convert any CPNS PGA entity to and from its DQGA entity form, all without changing orientation of the entities. Thus, each of the three algebras within PGA can be taken advantage of for what it does best, made possible by the operation J_e and identities relating CPNS PGA to DQGA. PGA G(3,0,1) is then doubled into a Double PGA (DPGA) G(6,0,2) including a Double DQGA (DDQGA), which feature two closely related forms of a general quadric entity that can be rotated, translated, and intersected with planes and lines. The paper then concludes with final remarks.