At the close New Zealand were 331 for five in their first innings, trailing the home side by 156.
Ryder (103) and Williamson (87 not out) defied the Indian bowlers after New Zealand had been reduced to 137 for four in reply to India's first innings 487.
Left-hander Ryder, who had Ross Taylor as a runner during the later part of his innings, brought up his third test century, each of them against India, with a cover-driven boundary off Shanthakumaran Sreesanth.
However, two deliveries later Sreesanth got his revenge when he made the ball straighten after pitching and caught Ryder plumb in front of the stumps.
The burly Ryder had grown in confidence as his innings progressed after surviving a chance on 11 when Rahul Dravid dropped him at a wide slip position off Sreesanth.
Williamson, 20, was particularly impressive with his footwork against the Indian spinners but he could consider himself lucky to survive a caught behind appeal on 56 against left-arm seamer Zaheer Khan.
Television replays showed a healthy edge and deviation as the ball passed the bat but debutant test umpire Kumar Dharmasena was unmoved.
India captain Mahendra Singh Dhoni utilised all his resources, including throwing the ball to Sachin Tendulkar to bowl his leg-spinners, but failed to break the partnership.
India's only other wickets in the day came before lunch when the spinners dismissed Brendon McCullum and Ross Taylor after their century stand for the third wicket.
McCullum (65) and Taylor (56), the most accomplished of the New Zealand batsmen, stitched together a 104-run partnership before the latter drove Harbhajan Singh straight to Vangipurappu Laxman at short midwicket.
McCullum had looked solid, using his feet well against the spinners during his 122-ball stay at the wicket, and hit 11 boundaries in his knock.
However, he dragged his feet out trying to defend a left-arm spinner Pragyan Ojha's delivery that spun across his bat. Dhoni made no mistake behind the stumps.
