A.
that is a transformation of the plane that carries any directional lines
Run-On Sentence: "that is" is an abridged version of "that is to say". Remember sentences starting with "that is" are independent clauses. Therefore, in this option, two independent clauses are joined by a comma, resulting in a run-on sentence. Had the sentence used a semi-colon, it would have been correct.
Style and Usage: "That is" is common'ly used in informal writing but would be considered unacceptable in more formal writing.
B.
transformation of the plane that carries any directional linesC.
that is, a transformation of the plane
which carries any directional lines
Run-On Sentence: "that is" is an abridged version of "that is to say". Remember sentences starting with "that is" are independent clauses. Therefore, in this option, two independent clauses are joined by a comma, resulting in a run-on sentence.
Style and Usage: "That is" is common'ly used in informal writing but would be considered unacceptable in more formal writing.
Modifier Error: Relative pronoun which MUST always be preceded by a comma.
D.
which is a transformation of the plane that carries any directional lines
Modifier Error: Relative pronoun which modifies the immediately preceding noun before the comma. Hence, the entire relative clause starting with which erroneously modifies the Euclidean plane.
E.
or a transformation of the plane
carrying any directional linesModifier Error: Verbing modifier without comma modifies the immediately preceding noun. Hence, the verbing phrase "carrying any directional lines" erroneously modifies plane.
Meaning Error: The usage of conjunction or distorts to the meaning of the sentence. In other words, the sentence as written provides two disjoint definitions of plane symmetry.