to develop a curved surface on a plane

Develop \De*vel"op\ (d[-e]*v[e^]l"[o^]p), v. t. [imp. & p. p.
   {Developed}; p. pr. & vb. n. {Developing}.] [F. d['e]veloper;
   d['e]- (L. dis-) + OF. voluper, voleper, to envelop, perh.
   from L. volup agreeably, delightfully, and hence orig., to
   make agreeable or comfortable by enveloping, to keep snug
   (cf. {Voluptuous}); or. perh. fr. a derivative of volvere,
   volutum, to roll (cf. {Devolve}). Cf. {Envelop}.] [Written
   also {develope}.]
   1. To free from that which infolds or envelops; to unfold; to
      lay open by degrees or in detail; to make visible or
      known; to disclose; to produce or give forth; as, to
      develop theories; a motor that develops 100 horse power.
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            These serve to develop its tenets.    --Milner.
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            The 20th was spent in strengthening our position and
            developing the line of the enemy.     --The Century.
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   2. To unfold gradually, as a flower from a bud; hence, to
      bring through a succession of states or stages, each of
      which is preparatory to the next; to form or expand by a
      process of growth; to cause to change gradually from an
      embryo, or a lower state, to a higher state or form of
      being; as, sunshine and rain develop the bud into a
      flower; to develop the mind.
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            The sound developed itself into a real compound.
                                                  --J. Peile.
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            All insects . . . acquire the jointed legs before
            the wings are fully developed.        --Owen.
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   3. To advance; to further; to prefect; to make to increase;
      to promote the growth of.
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            We must develop our own resources to the utmost.
                                                  --Jowett
                                                  (Thucyd).
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   4. (Math.) To change the form of, as of an algebraic
      expression, by executing certain indicated operations
      without changing the value.
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   5. (Photog.) To cause to become visible, as an invisible or
      latent image upon plate, by submitting it to chemical
      agents; to bring to view.
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   {To develop a curved surface on a plane} (Geom.), to produce
      on the plane an equivalent surface, as if by rolling the
      curved surface so that all parts shall successively touch
      the plane.

   Syn: To uncover; unfold; evolve; promote; project; lay open;
        disclose; exhibit; unravel; disentangle.
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